16144:
15927:
16165:
16133:
16202:
16175:
16155:
8355:
8050:
4300:
3829:
12319:
15544:
13917:
4546:
13141:
1873:
95:
associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different.
4118:
3620:
12042:
6442:
15288:
2130:
7977:
2994:
6632:
13648:
8350:{\displaystyle {\begin{aligned}c_{t}(\operatorname {Sym} ^{p}E)&=\prod _{i_{1}\leq \cdots \leq i_{p}}(1+(\alpha _{i_{1}}+\cdots +\alpha _{i_{p}})t),\\c_{t}(\wedge ^{p}E)&=\prod _{i_{1}<\cdots <i_{p}}(1+(\alpha _{i_{1}}+\cdots +\alpha _{i_{p}})t).\end{aligned}}}
11617:
5981:
14493:. The formal properties of the Chern classes remain the same, with one crucial difference: the rule which computes the first Chern class of a tensor product of line bundles in terms of first Chern classes of the factors is not (ordinary) addition, but rather a
13657:
12827:
6248:
14950:
4367:
249:
vector bundle (the "hairs" on a ball are actually copies of the real line), there are generalizations in which the hairs are complex (see the example of the complex hairy ball theorem below), or for 1-dimensional projective spaces over many other fields.
12700:
11804:
12832:
13370:
8831:
7263:
2800:
6261:
14193:
14120:
12006:
3266:
1747:
3477:
7720:
702:
8636:
9041:
2863:
10748:
6460:
1955:
13488:
7643:
4113:
211:
Chern's approach used differential geometry, via the curvature approach described predominantly in this article. He showed that the earlier definition was in fact equivalent to his. The resulting theory is known as the
4889:
9750:
2606:
14543:) reducing degrees by 1. This corresponds to the fact that the Chow groups are a sort of analog of homology groups, and elements of cohomology groups can be thought of as homomorphisms of homology groups using the
12479:
4295:{\displaystyle T\mathbb {C} \mathbb {P} ^{n}\oplus {\mathcal {O}}=\operatorname {Hom} ({\mathcal {O}}(-1),\eta )\oplus \operatorname {Hom} ({\mathcal {O}}(-1),{\mathcal {O}}(-1))={\mathcal {O}}(1)^{\oplus (n+1)}.}
3824:{\displaystyle 0\to \Omega _{\mathbb {C} \mathbb {P} ^{n}}|_{U}{\overset {dz_{i}\mapsto e_{i}}{\to }}\oplus _{1}^{n+1}{\mathcal {O}}(-1)|_{U}{\overset {e_{i}\mapsto z_{i}}{\to }}{\mathcal {O}}_{U}\to 0,\,i\geq 0,}
11507:
6068:
10451:
5812:
12314:{\displaystyle {\begin{aligned}c({\mathcal {T}}_{X})&={\frac {1+5h+10h^{2}+10h^{3}}{1+5h}}\\&=\left(1+5h+10h^{2}+10h^{3}\right)\left(1-5h+25h^{2}-125h^{3}\right)\\&=1+10h^{2}-40h^{3}\end{aligned}}}
4665:
1501:
is connected. Hence, one simply defines the top Chern class of the bundle to be its Euler class (the Euler class of the underlying real vector bundle) and handles lower Chern classes in an inductive fashion.
6852:
4608:
459:
5653:
9859:
5280:
4039:
2361:
15539:{\displaystyle {\begin{aligned}c(V)&=c(L_{1}\oplus \cdots \oplus L_{n})\\&=\prod _{i=1}^{n}c(L_{i})\\&=\prod _{i=1}^{n}(1+x_{i})\\&=\sum _{i=0}^{n}e_{i}(x_{1},\ldots ,x_{n})\end{aligned}}}
8944:
5024:
3533:
2504:
12705:
5442:
234:) over any nonsingular variety. Algebro-geometric Chern classes do not require the underlying field to have any special properties. In particular, the vector bundles need not necessarily be complex.
7039:
114:
Chern classes are also feasible to calculate in practice. In differential geometry (and some types of algebraic geometry), the Chern classes can be expressed as polynomials in the coefficients of the
9317:
6964:
6077:
3329:
14802:
11212:
9937:
12598:
11702:
10010:
7367:
3089:
1499:
15208:
15293:
12837:
12047:
8055:
1049:
3037:
1742:
976:
13424:
5332:
468:
In algebraic geometry, this classification of (isomorphism classes of) complex line bundles by the first Chern class is a crude approximation to the classification of (isomorphism classes of)
7715:
6925:
5172:
11278:
822:
15283:
13242:
8025:
11459:
11156:
9634:
8426:
7509:
3982:
1440:
14033:
3937:
9505:
7104:
10080:
9096:
12573:
12368:
11502:
2682:
10582:
9321:
Since the values are in integral cohomology groups, rather than cohomology with real coefficients, these Chern classes are slightly more refined than those in the
Riemannian example.
9202:
4362:
10502:
7086:
4733:
11697:
11358:
11064:
5507:
2278:
360:
14124:
10977:
2244:
1587:
14051:
13483:
11867:
10347:
10118:
3193:
3155:
1915:
3899:
3564:
3335:
2850:
1623:
9788:
8882:
2184:
1535:
13912:{\displaystyle \operatorname {ch} (V)=\operatorname {rk} (V)+c_{1}(V)+{\frac {1}{2}}(c_{1}(V)^{2}-2c_{2}(V))+{\frac {1}{6}}(c_{1}(V)^{3}-3c_{1}(V)c_{2}(V)+3c_{3}(V))+\cdots .}
12037:
11657:
10309:
6716:
2405:
11862:
10244:
10164:
8669:
8477:
3615:
11395:
2675:
606:
15069:
14396:
11321:
11022:
10812:
9372:
4541:{\displaystyle c(\mathbb {C} \mathbb {P} ^{n}){\overset {\mathrm {def} }{=}}c(T\mathbb {CP} ^{n})=c({\mathcal {O}}_{\mathbb {C} \mathbb {P} ^{n}}(1))^{n+1}=(1+a)^{n+1},}
169:
can therefore be defined as the pullback of the Chern classes of the universal bundle. In turn, these universal Chern classes can be explicitly written down in terms of
14359:
9566:
1950:
730:
14046:
The Chern character is useful in part because it facilitates the computation of the Chern class of a tensor product. Specifically, it obeys the following identities:
10863:
10132:
that the total Chern class of an arbitrary finite rank complex vector bundle can be defined in terms of the first Chern class of a tautologically-defined line bundle.
9411:
5749:
5543:
5079:
1091:
564:
10617:
4793:
1680:
761:
595:
10919:
10612:
10206:
9537:
15121:
15042:
14505:
In algebraic geometry there is a similar theory of Chern classes of vector bundles. There are several variations depending on what groups the Chern classes lie in:
14423:
13136:{\displaystyle {\begin{aligned}c({\mathcal {T}}_{X})&={\frac {(1+)^{4}}{(1+d)}}\\&=(1+4+6^{2})(1-d+d^{2}^{2})\\&=1+(4-d)+(6-4d+d^{2})^{2}\end{aligned}}}
12506:
10264:
7405:
4760:
3857:
15547:
13187:
8664:
8506:
7521:
6071:
3198:
14446:
14299:
14276:
13210:
4798:
14470:
14319:
13161:
12593:
12530:
11826:
11469:
Computing the characteristic classes for projective space forms the basis for many characteristic class computations since for any smooth projective subvariety
11084:
9639:
8045:
7425:
2511:
14475:
The Chern numbers of the tangent bundle of a complex (or almost complex) manifold are called the Chern numbers of the manifold, and are important invariants.
12377:
10752:
One then may check that this alternative definition coincides with whatever other definition one may favor, or use the previous axiomatic characterization.
8511:
3859:
are the basis of the middle term. The same sequence is clearly then exact on the whole projective space and the dual of it is the aforementioned sequence.
10359:
8949:
6437:{\displaystyle \operatorname {ch} (E)=\operatorname {rk} +c_{1}+{\frac {1}{2}}(c_{1}^{2}-2c_{2})+{\frac {1}{6}}(c_{1}^{3}-3c_{1}c_{2}+3c_{3})+\cdots .}
16205:
6749:
5556:
1868:{\displaystyle \cdots \to \operatorname {H} ^{k}(B;\mathbb {Z} ){\overset {\pi |_{B'}^{*}}{\to }}\operatorname {H} ^{k}(B';\mathbb {Z} )\to \cdots ,}
3990:
2304:
4044:
4921:
2415:
7972:{\displaystyle c_{1}(E)+rc_{1}(L),\dots ,\sum _{j=0}^{i}{\binom {r-i+j}{j}}c_{i-j}(E)c_{1}(L)^{j},\dots ,\sum _{j=0}^{r}c_{r-j}(E)c_{1}(L)^{j}.}
5337:
1449:
One can define a Chern class in terms of an Euler class. This is the approach in the book by Milnor and
Stasheff, and emphasizes the role of an
9218:
4898:
A Chern polynomial is a convenient way to handle Chern classes and related notions systematically. By definition, for a complex vector bundle
365:
14197:
As stated above, using the
Grothendieck additivity axiom for Chern classes, the first of these identities can be generalized to state that
5986:
5189:
2989:{\displaystyle 0\to {\mathcal {O}}_{\mathbb {CP} ^{n}}\to {\mathcal {O}}_{\mathbb {CP} ^{n}}(1)^{\oplus (n+1)}\to T\mathbb {CP} ^{n}\to 0}
6627:{\displaystyle \operatorname {td} (E)=\prod _{1}^{n}{a_{i} \over 1-e^{-a_{i}}}=1+{1 \over 2}c_{1}+{1 \over 12}(c_{1}^{2}+c_{2})+\cdots .}
4613:
2125:{\displaystyle c_{k}(E)={\begin{cases}{\pi |_{B'}^{*}}^{-1}c_{k}(E')&k<n\\e(E_{\mathbb {R} })&k=n\\0&k>n\end{cases}}}
13643:{\displaystyle \operatorname {ch} (V)=e^{x_{1}}+\cdots +e^{x_{n}}:=\sum _{m=0}^{\infty }{\frac {1}{m!}}(x_{1}^{m}+\cdots +x_{n}^{m}).}
6857:
4555:
16231:
770:
15213:
9808:
12010:
Since the Chow ring of a hypersurface is difficult to compute, we will consider this sequence as a sequence of coherent sheaves in
9571:
8887:
1098:
362:, which associates to a line bundle its first Chern class. Moreover, this bijection is a group homomorphism (thus an isomorphism):
13943:
6639:: The observation that a Chern class is essentially an elementary symmetric polynomial can be used to "define" Chern classes. Let
14329:
of the vector bundle. For example, if the manifold has dimension 6, there are three linearly independent Chern numbers, given by
14240:
3485:
6985:
10891:
There is another construction of Chern classes which take values in the algebrogeometric analogue of the cohomology ring, the
6935:
3271:
15783:
15765:
15716:
15690:
15624:
11612:{\displaystyle 0\to {\mathcal {T}}_{X}\to {\mathcal {T}}_{\mathbb {P} ^{n}}|_{X}\to {\mathcal {N}}_{X/\mathbb {P} ^{n}}\to 0}
9157:
5976:{\displaystyle t_{1}^{k}+\cdots +t_{n}^{k}=s_{k}(\sigma _{1}(t_{1},\ldots ,t_{n}),\ldots ,\sigma _{k}(t_{1},\ldots ,t_{n}))}
10349:
to minus the (Poincaré-dual) class of the hyperplane, that spans the cohomology of the fiber, in view of the cohomology of
7283:
3042:
1463:
15126:
981:
2999:
908:
126:
There are various ways of approaching the subject, each of which focuses on a slightly different flavor of Chern class.
15839:
13377:
10895:. It can be shown that there is a unique theory of Chern classes such that if you are given an algebraic vector bundle
5285:
295:. Then the only nontrivial Chern class is the first Chern class, which is an element of the second cohomology group of
28:
10269:
7648:
6675:
5092:
2368:
16193:
16188:
15740:
14729:
15730:
7984:
15093:
12822:{\displaystyle c({\mathcal {T}}_{X})={\frac {c({\mathcal {T}}_{\mathbb {P} ^{3}|_{X}})}{c({\mathcal {O}}_{X}(d))}}}
11400:
11093:
8360:
7430:
5662:
3950:
2679:
We must show that this cohomology class is non-zero. It suffices to compute its integral over the
Riemann sphere:
3908:
2613:
16226:
16183:
10453:
therefore form a family of ambient cohomology classes restricting to a basis of the cohomology of the fiber. The
9432:
6243:{\displaystyle \operatorname {ch} (E)=e^{a_{1}(E)}+\cdots +e^{a_{n}(E)}=\sum s_{k}(c_{1}(E),\ldots ,c_{n}(E))/k!}
108:
14945:{\displaystyle H^{2*}(M,\mathbb {Z} )\to \oplus _{k}^{\infty }\eta (H^{2*}(M,\mathbb {Z} )),x\mapsto xt^{|x|/2}}
10026:
9046:
237:
Regardless of the particular paradigm, the intuitive meaning of the Chern class concerns 'required zeroes' of a
12695:{\displaystyle 0\to {\mathcal {T}}_{X}\to {\mathcal {T}}_{\mathbb {P} ^{3}}|_{X}\to {\mathcal {O}}_{X}(d)\to 0}
12543:
12326:
11799:{\displaystyle 0\to {\mathcal {T}}_{X}\to {\mathcal {T}}_{\mathbb {P} ^{4}}|_{X}\to {\mathcal {O}}_{X}(5)\to 0}
11472:
9414:
1450:
898:. It can be shown that the cohomology classes of the Chern forms do not depend on the choice of connection in
11163:
10526:
9888:
1693:
1592:
16085:
14484:
9946:
4309:
230:. The generalized Chern classes in algebraic geometry can be defined for vector bundles (or more precisely,
103:
sections a vector bundle has. The Chern classes offer some information about this through, for instance, the
10460:
7044:
4670:
1505:
The precise construction is as follows. The idea is to do base change to get a bundle of one-less rank. Let
11662:
11326:
11029:
9540:
5447:
2249:
477:
321:
14952:
where the left is the cohomology ring of even terms, η is a ring homomorphism that disregards grading and
10924:
2365:
This is evinced by the fact that a trivial bundle always admits a flat connection. So, we shall show that
2208:
483:
For complex vector bundles of dimension greater than one, the Chern classes are not a complete invariant.
13429:
10314:
10085:
6731:
is exactly the ring of symmetric polynomials, which are polynomials in elementary symmetric polynomials Ï
3160:
3114:
3869:
3538:
2820:
16093:
14489:
14483:
There is a generalization of the theory of Chern classes, where ordinary cohomology is replaced with a
14322:
13365:{\displaystyle \operatorname {ch} (L)=\exp(c_{1}(L)):=\sum _{m=0}^{\infty }{\frac {c_{1}(L)^{m}}{m!}}.}
9761:
8855:
8826:{\displaystyle c(\operatorname {Sym} ^{2}E)=1+4c_{1}+5c_{1}^{2}+5c_{2}+2c_{1}^{3}+11c_{1}c_{2}+7c_{3}.}
2157:
1508:
307:
266:
15760:, Annals of Mathematics Studies, vol. 76, Princeton University Press; University of Tokyo Press,
12013:
11633:
11217:
14672:
11831:
10218:
10138:
3092:
1555:
571:
238:
14512:
For varieties over general fields, the Chern classes can take values in cohomology theories such as
8433:
7258:{\displaystyle \operatorname {Nat} (,H^{*}(-,\mathbb {Z} ))=H^{*}(G_{n},\mathbb {Z} )=\mathbb {Z} .}
3569:
1986:
1878:
99:
In topology, differential geometry, and algebraic geometry, it is often important to count how many
15892:
11365:
10760:
In fact, these properties uniquely characterize the Chern classes. They imply, among other things:
10454:
10350:
10129:
9755:
2191:
853:
104:
63:
50:. They have since become fundamental concepts in many branches of mathematics and physics, such as
2795:{\displaystyle \int c_{1}={\frac {i}{\pi }}\int {\frac {dz\wedge d{\bar {z}}}{(1+|z|^{2})^{2}}}=2}
16178:
16164:
15047:
14561:
8852:
We can use these abstract properties to compute the rest of the chern classes of line bundles on
887:
469:
254:
55:
14364:
12532:
by using the definition of the Euler characteristic and using the
Lefschetz hyperplane theorem.
12512:). The Euler characteristic can then be used to compute the Betti numbers for the cohomology of
11284:
10985:
10775:
9335:
2134:
It then takes some work to check the axioms of Chern classes are satisfied for this definition.
16113:
16034:
15911:
15899:
15872:
15832:
15634:
14332:
9871:
9545:
2301:
For this, we need the following fact: the first Chern class of a trivial bundle is zero, i.e.,
1920:
709:
220:
129:
The original approach to Chern classes was via algebraic topology: the Chern classes arise via
16108:
14642:
vector bundles, by the intermediation of compatible almost complex structures. In particular,
12509:
10832:
9384:
5718:
5512:
5048:
1058:
533:
15955:
15882:
15577:
14697:
14224:. The second identity establishes the fact that this homomorphism also respects products in
14188:{\displaystyle \operatorname {ch} (V\otimes W)=\operatorname {ch} (V)\operatorname {ch} (W).}
13937:
9107:
6649:
4772:
1457:
746:
580:
497:
213:
92:
44:
14115:{\displaystyle \operatorname {ch} (V\oplus W)=\operatorname {ch} (V)+\operatorname {ch} (W)}
12001:{\displaystyle c({\mathcal {T}}_{X})c({\mathcal {O}}_{X}(5))=(1+h)^{5}=1+5h+10h^{2}+10h^{3}}
10898:
10587:
10185:
9516:
6932:: Any characteristic class is a polynomial in Chern classes, for the reason as follows. Let
3261:{\displaystyle \pi \colon \mathbb {C} ^{n+1}\setminus \{0\}\to \mathbb {C} \mathbb {P} ^{n}}
2138:
16103:
16055:
16029:
15877:
15672:
15099:
15020:
14401:
13936:
If a connection is used to define the Chern classes when the base is a manifold (i.e., the
13230:
12484:
10249:
7383:
6979:
5690:
4738:
3835:
3472:{\displaystyle \pi ^{*}d(z_{i}/z_{0})={z_{0}dz_{i}-z_{i}dz_{0} \over z_{0}^{2}},\,i\geq 1.}
2199:
231:
100:
88:
40:
8:
15950:
14692:
14643:
14639:
14509:
For complex varieties the Chern classes can take values in ordinary cohomology, as above.
14252:
13922:
13166:
8643:
8485:
5712:
3902:
857:
833:
310:
with which to classify complex line bundles, topologically speaking. That is, there is a
16154:
15798:
14428:
14281:
14258:
13192:
9799:
5755:
need not be a direct sum of line bundles in the preceding discussion. The conclusion is
1660:
277:
16148:
16118:
16098:
16019:
16009:
15887:
15867:
15602:
14455:
14304:
13146:
12578:
12515:
11811:
11069:
8030:
7410:
6719:
2807:
2290:
895:
473:
242:
227:
32:
24:
5334:
is a direct sum of (complex) line bundles, then it follows from the sum formula that:
16143:
16136:
16002:
15960:
15825:
15779:
15761:
15736:
15712:
15686:
15660:
15638:
15620:
15594:
15572:
14725:
14687:
14655:
14604:
14517:
14449:
11627:
9114:
7089:
7088:
Characteristic classes form a ring because of the ring structure of cohomology ring.
6657:
3096:
2803:
1538:
697:{\displaystyle \det \left({\frac {it\Omega }{2\pi }}+I\right)=\sum _{k}c_{k}(V)t^{k}}
241:
of a vector bundle: for example the theorem saying one can't comb a hairy ball flat (
134:
67:
16168:
14623:âČ is another almost complex manifold of the same dimension, then it is cobordant to
8631:{\displaystyle c(\operatorname {Sym} ^{2}E)=1+3c_{1}+2c_{1}^{2}+4c_{2}+4c_{1}c_{2},}
465:
of complex line bundles corresponds to the addition in the second cohomology group.
15916:
15862:
15808:
15650:
15586:
14749:
14667:
14513:
14494:
9036:{\displaystyle c_{1}({\mathcal {O}}(1))=1\in H^{2}(\mathbb {CP} ^{1};\mathbb {Z} )}
8840:
5177:
The
Whitney sum formula, one of the axioms of Chern classes (see below), says that
891:
736:
with coefficients in the commutative algebra of even complex differential forms on
503:
14765:
Editorial note: Our notation differs from MilnorâStasheff, but seems more natural.
2410:
15975:
15970:
15708:
15680:
15668:
15614:
11087:
9510:
9422:
3480:
872:
825:
520:
130:
16158:
10743:{\displaystyle -a^{n}=c_{1}(E)\cdot a^{n-1}+\cdots +c_{n-1}(E)\cdot a+c_{n}(E).}
16065:
15997:
14775:
14572:
14527:
over general fields the Chern classes can also take values in homomorphisms of
14040:
9940:
2285:
2187:
1687:
741:
575:
462:
170:
115:
15700:
15017:
Use, for example, WolframAlpha to expand the polynomial and then use the fact
4610:; i.e., the negative of the first Chern class of the tautological line bundle
16220:
16075:
15985:
15965:
15802:
15664:
15598:
14745:
14588:
14206:
13233:
of a space to (the completion of) its rational cohomology. For a line bundle
13213:
3100:
1052:
513:
506:
165:, of a universal bundle over the classifying space, and the Chern classes of
51:
47:
15812:
7638:{\displaystyle c_{t}(E\otimes L)=\sum _{i=0}^{r}c_{i}(E)c_{t}(L)^{r-i}t^{i}}
4108:{\displaystyle {\mathcal {O}}(-1)\oplus \eta ={\mathcal {O}}^{\oplus (n+1)}}
16060:
15980:
15926:
15753:
15726:
14202:
7092:
says this ring of characteristic classes is exactly the cohomology ring of
6970:, assigns the set of isomorphism classes of complex vector bundles of rank
4884:{\displaystyle c_{k}(\mathbb {C} \mathbb {P} ^{n})={\binom {n+1}{k}}a^{k}.}
138:
14531:
CH(V): for example, the first Chern class of a line bundle over a variety
9745:{\displaystyle c_{k}(E\oplus F)=\sum _{i=0}^{k}c_{i}(E)\smile c_{k-i}(F).}
15749:
14682:
14677:
14544:
12371:
10874:
10121:
10020:
7378:
2601:{\displaystyle \Omega ={\frac {2dz\wedge d{\bar {z}}}{(1+|z|^{2})^{2}}}.}
841:
837:
300:
292:
59:
20:
14301:(i.e., the sum of indices of the Chern classes in the product should be
13229:
Chern classes can be used to construct a homomorphism of rings from the
16014:
15945:
15904:
15655:
15606:
14717:
14528:
12474:{\displaystyle \int _{}c_{3}({\mathcal {T}}_{X})=\int _{}-40h^{3}=-200}
9803:
9125:
9043:. Then using tensor powers, we can relate them to the chern classes of
6450:
3985:
2811:
281:
12370:
to compute the Euler characteristic. Traditionally this is called the
10246:
is equipped with its tautological complex line bundle, that we denote
223:
showing that axiomatically one need only define the line bundle case.
16039:
14557:
14487:. The theories for which such generalization is possible are called
10892:
311:
15590:
14583:
are thus defined to be the Chern classes of its tangent bundle. If
14448:, the number of possible independent Chern numbers is the number of
6063:{\displaystyle s_{1}=\sigma _{1},s_{2}=\sigma _{1}^{2}-2\sigma _{2}}
5751:
factorizes into linear factors after enlarging the cohomology ring;
16024:
15992:
15941:
15848:
14799:
In a ring-theoretic term, there is an isomorphism of graded rings:
14596:
14210:
10446:{\displaystyle 1,a,a^{2},\ldots ,a^{n-1}\in H^{*}(\mathbb {P} (E))}
2202:
2149:
9209:
4660:{\displaystyle {\mathcal {O}}_{\mathbb {C} \mathbb {P} ^{n}}(-1)}
14724:(Corr. 3. print. ed.). New York : Springer. p. 267ff.
10921:
over a quasi-projective variety there are a sequence of classes
9539:
is another complex vector bundle, then the Chern classes of the
6847:{\displaystyle f_{E}^{*}:\mathbb {Z} \to H^{*}(X,\mathbb {Z} ).}
4603:{\displaystyle H^{2}(\mathbb {C} \mathbb {P} ^{n},\mathbb {Z} )}
192:, the maps must be homotopic. Therefore, the pullback by either
14649:
5648:{\displaystyle c_{k}(E)=\sigma _{k}(a_{1}(E),\ldots ,a_{n}(E))}
15722:(Provides a very short, introductory review of Chern classes).
9854:{\displaystyle \mathbb {CP} ^{k-1}\subseteq \mathbb {CP} ^{k}}
4034:{\displaystyle \operatorname {Hom} ({\mathcal {O}}(-1),\eta )}
2356:{\displaystyle c_{1}(\mathbb {CP} ^{1}\times \mathbb {C} )=0.}
204:
must be the same class. This shows that the Chern classes of
133:
which provides a mapping associated with a vector bundle to a
8939:{\displaystyle {\mathcal {O}}(-1)^{*}\cong {\mathcal {O}}(1)}
883:
188:
to the classifying space whose pullbacks are the same bundle
15817:
5019:{\displaystyle c_{t}(E)=1+c_{1}(E)t+\cdots +c_{n}(E)t^{n}.}
3528:{\displaystyle \Omega _{\mathbb {C} \mathbb {P} ^{n}}|_{U}}
2499:{\displaystyle h={\frac {dzd{\bar {z}}}{(1+|z|^{2})^{2}}}.}
2118:
14325:(or "integrated over the manifold") to give an integer, a
13426:
is a direct sum of line bundles, with first Chern classes
12323:
Using the Gauss-Bonnet theorem we can integrate the class
10504:
can be written uniquely as a linear combination of the 1,
5702:'s is a polynomial in elementary symmetric polynomials in
5437:{\displaystyle c_{t}(E)=(1+a_{1}(E)t)\cdots (1+a_{n}(E)t)}
2814:
would integrate to 0, so the cohomology class is nonzero.
245:). Although that is strictly speaking a question about a
15575:(1946), "Characteristic classes of Hermitian Manifolds",
13143:
Giving the total chern class. In particular, we can find
7034:{\displaystyle \operatorname {Vect} _{n}^{\mathbb {C} }=}
2246:
be the bundle of complex tangent vectors having the form
9878:) replaced these with a slightly smaller set of axioms:
9312:{\displaystyle c(E)=c_{0}(E)+c_{1}(E)+c_{2}(E)+\cdots .}
6959:{\displaystyle \operatorname {Vect} _{n}^{\mathbb {C} }}
3324:{\displaystyle U=\mathbb {CP} ^{n}\setminus \{z_{0}=0\}}
3039:
is the structure sheaf (i.e., the trivial line bundle),
2610:
Furthermore, by the definition of the first Chern class
2508:
One readily shows that the curvature 2-form is given by
1460:
comes with a canonical orientation, ultimately because
1093:, we get the following expression for the Chern forms:
12595:
smooth hypersurface, we have the short exact sequence
5553:, determine the coefficients of the polynomial: i.e.,
454:{\displaystyle c_{1}(L\otimes L')=c_{1}(L)+c_{1}(L');}
200:
of any universal Chern class to a cohomology class of
15291:
15216:
15129:
15102:
15050:
15023:
14805:
14458:
14431:
14404:
14367:
14335:
14307:
14284:
14261:
14127:
14054:
13946:
13660:
13491:
13432:
13380:
13245:
13195:
13169:
13149:
12835:
12708:
12601:
12581:
12546:
12518:
12487:
12380:
12329:
12045:
12016:
11870:
11834:
11814:
11705:
11665:
11636:
11510:
11475:
11403:
11368:
11329:
11287:
11220:
11166:
11096:
11072:
11032:
10988:
10927:
10901:
10835:
10778:
10620:
10590:
10529:
10463:
10362:
10317:
10272:
10252:
10221:
10188:
10141:
10088:
10029:
9949:
9891:
9811:
9764:
9642:
9574:
9548:
9519:
9435:
9387:
9338:
9329:
The Chern classes satisfy the following four axioms:
9221:
9160:
9049:
8952:
8890:
8858:
8672:
8646:
8514:
8488:
8436:
8363:
8053:
8033:
7987:
7723:
7651:
7524:
7433:
7413:
7386:
7362:{\displaystyle c_{t}(E)=\sum _{i=0}^{r}c_{i}(E)t^{i}}
7286:
7107:
7047:
6988:
6938:
6860:
6752:
6678:
6463:
6264:
6080:
5989:
5815:
5721:
5559:
5515:
5450:
5340:
5288:
5192:
5095:
5051:
4924:
4801:
4775:
4741:
4673:
4616:
4558:
4370:
4312:
4121:
4047:
3993:
3953:
3911:
3901:
that passes through the origin. It is an exercise in
3872:
3838:
3623:
3572:
3541:
3488:
3338:
3274:
3201:
3163:
3117:
3084:{\displaystyle {\mathcal {O}}_{\mathbb {CP} ^{n}}(1)}
3045:
3002:
2866:
2823:
2685:
2616:
2514:
2418:
2371:
2307:
2252:
2211:
2160:
1958:
1923:
1881:
1750:
1696:
1663:
1595:
1558:
1511:
1494:{\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}
1466:
1101:
1061:
984:
911:
882:
of the Chern class indicates that 'class' here means
773:
749:
712:
609:
583:
536:
526:, representatives of each Chern class (also called a
368:
324:
314:
between the isomorphism classes of line bundles over
15203:{\displaystyle c_{i}(V)=e_{i}(x_{1},\ldots ,x_{n}).}
14278:, then any product of Chern classes of total degree
13940:), then the explicit form of the Chern character is
5275:{\displaystyle c_{t}(E\oplus E')=c_{t}(E)c_{t}(E').}
10519:In particular, one may define the Chern classes of
6966:be the contravariant functor that, to a CW complex
4552:is the canonical generator of the cohomology group
1044:{\displaystyle \det(X)=\exp(\mathrm {tr} (\ln(X)))}
15805:. Contains a chapter about characteristic classes.
15538:
15277:
15202:
15115:
15063:
15036:
14944:
14478:
14464:
14440:
14417:
14390:
14353:
14313:
14293:
14270:
14187:
14114:
14027:
13911:
13642:
13477:
13418:
13364:
13204:
13181:
13155:
13135:
12821:
12694:
12587:
12567:
12524:
12500:
12473:
12362:
12313:
12031:
12000:
11856:
11820:
11798:
11691:
11651:
11611:
11496:
11453:
11389:
11352:
11315:
11272:
11206:
11150:
11078:
11058:
11016:
10971:
10913:
10857:
10806:
10742:
10606:
10576:
10496:
10445:
10341:
10303:
10258:
10238:
10200:
10158:
10112:
10074:
10004:
9931:
9865:
9853:
9782:
9744:
9628:
9560:
9531:
9499:
9405:
9366:
9311:
9196:
9090:
9035:
8938:
8876:
8825:
8658:
8630:
8500:
8471:
8420:
8349:
8039:
8019:
7971:
7709:
7637:
7503:
7419:
7399:
7361:
7257:
7080:
7033:
6958:
6919:
6846:
6710:
6626:
6436:
6242:
6062:
5975:
5743:
5647:
5537:
5501:
5436:
5326:
5274:
5166:
5073:
5018:
4883:
4787:
4754:
4727:
4659:
4602:
4540:
4356:
4294:
4107:
4033:
3976:
3931:
3893:
3851:
3823:
3609:
3558:
3527:
3471:
3323:
3260:
3187:
3149:
3083:
3032:{\displaystyle {\mathcal {O}}_{\mathbb {CP} ^{n}}}
3031:
2988:
2844:
2794:
2669:
2600:
2498:
2399:
2355:
2272:
2238:
2178:
2124:
1944:
1909:
1867:
1736:
1674:
1617:
1581:
1529:
1493:
1434:
1085:
1043:
971:{\displaystyle \mathrm {tr} (\ln(X))=\ln(\det(X))}
970:
816:
755:
724:
696:
589:
558:
453:
354:
260:
13419:{\displaystyle V=L_{1}\oplus \cdots \oplus L_{n}}
9324:
7828:
7801:
6660:in the sense that, given a complex vector bundle
5327:{\displaystyle E=L_{1}\oplus \cdots \oplus L_{n}}
5028:This is not a new invariant: the formal variable
4862:
4841:
16218:
15554:) above solely in terms of the Chern classes of
13921:This last expression, justified by invoking the
13189:is even, so every smooth hypersurface of degree
7710:{\displaystyle c_{i}(E\otimes L),i=1,2,\dots ,r}
6920:{\displaystyle c_{k}(E)=f_{E}^{*}(\sigma _{k}).}
5167:{\displaystyle c(E)=1+c_{1}(E)+\cdots +c_{n}(E)}
2150:The complex tangent bundle of the Riemann sphere
1625:such that each fiber is the quotient of a fiber
985:
953:
610:
15550:may be used to re-express the power sums in ch(
15088:is a sum of line bundles, the Chern classes of
2860:There is an exact sequence of sheaves/bundles:
817:{\displaystyle \Omega =d\omega +{\frac {1}{2}}}
15748:
15278:{\displaystyle c(V):=\sum _{i=0}^{n}c_{i}(V),}
8020:{\displaystyle \alpha _{1},\dots ,\alpha _{r}}
6978:and, to a map, its pullback. By definition, a
3106:There are two ways to get the above sequence:
299:. As it is the top Chern class, it equals the
157:to the classifying space such that the bundle
15833:
15643:Bulletin de la Société Mathématique de France
14425:. In general, if the manifold has dimension
11864:. Then the Whitney sum formula gives us that
11454:{\displaystyle c:K_{0}(X)\to A^{\bullet }(X)}
11151:{\displaystyle c_{1}({\mathcal {O}}_{X}(D))=}
9629:{\displaystyle c(E\oplus F)=c(E)\smile c(F);}
8421:{\displaystyle c_{1}(\wedge ^{r}E)=c_{1}(E).}
7504:{\displaystyle c_{i}(E^{*})=(-1)^{i}c_{i}(E)}
3977:{\displaystyle T\mathbb {C} \mathbb {P} ^{n}}
1435:{\displaystyle \sum _{k}c_{k}(V)t^{k}=\left.}
836:, or via the same expression in which Ï is a
141:in this case). For any complex vector bundle
15633:
14650:Arithmetic schemes and Diophantine equations
14603:in the Chern classes can be paired with the
14028:{\displaystyle \operatorname {ch} (V)=\left}
9875:
9754:Normalization: The total Chern class of the
6656:-dimensional complex vector spaces. It is a
3947:to its complement. Thus, the tangent bundle
3932:{\displaystyle \mathbb {C} \mathbb {P} ^{n}}
3318:
3299:
3235:
3229:
2298:has no section which is everywhere nonzero.
491:
82:
14551:
9500:{\displaystyle c_{k}(f^{*}E)=f^{*}c_{k}(E)}
8847:
5758:"One can evaluate any symmetric polynomial
2855:
16201:
16174:
15840:
15826:
15705:Riemannian Geometry and Geometric Analysis
14556:The theory of Chern classes gives rise to
13485:the Chern character is defined additively
11160:Given an exact sequence of vector bundles
10516:with classes on the base as coefficients.
10075:{\displaystyle c(E)=1+e(E_{\mathbb {R} })}
9091:{\displaystyle c_{1}({\mathcal {O}}(n))=n}
5693:is that any symmetric polynomial in, say,
732:matrices whose entries are polynomials in
15685:. Springer Science & Business Media.
15654:
15619:. Springer Science & Business Media.
14883:
14829:
12755:
12636:
12568:{\displaystyle X\subset \mathbb {P} ^{3}}
12555:
12535:
12363:{\displaystyle c_{3}({\mathcal {T}}_{X})}
12019:
11740:
11639:
11591:
11545:
11497:{\displaystyle X\subset \mathbb {P} ^{n}}
11484:
10478:
10427:
10319:
10223:
10143:
10135:Namely, introducing the projectivization
10101:
10063:
9841:
9838:
9817:
9814:
9770:
9767:
9184:
9026:
9012:
9009:
8864:
8861:
7213:
7202:
7162:
7068:
7000:
6950:
6834:
6772:
4822:
4816:
4633:
4627:
4593:
4579:
4573:
4467:
4461:
4429:
4426:
4384:
4378:
4132:
4126:
3964:
3958:
3943:is naturally the set of linear maps from
3919:
3913:
3905:to see that the complex tangent space to
3875:
3808:
3642:
3636:
3501:
3495:
3459:
3286:
3283:
3248:
3242:
3210:
3166:
3060:
3057:
3017:
3014:
2970:
2967:
2916:
2913:
2887:
2884:
2832:
2829:
2340:
2326:
2323:
2226:
2223:
2166:
2163:
2077:
1849:
1780:
1484:
345:
15776:Algebraic Geometry, a concise dictionary
15044:are elementary symmetric polynomials in
14722:Differential forms in algebraic topology
14571:is an almost complex manifold, then its
11207:{\displaystyle 0\to E'\to E\to E''\to 0}
10886:
10881:
10822:. Thus the total Chern class terminates.
10577:{\displaystyle c_{1}(E),\ldots c_{n}(E)}
9932:{\displaystyle 0\to E'\to E\to E''\to 0}
5715:or by ring theory, any Chern polynomial
1737:{\displaystyle \pi |_{B'}\colon B'\to B}
1633:by the line spanned by a nonzero vector
306:The first Chern class turns out to be a
16:Characteristic classes of vector bundles
13237:, the Chern character ch is defined by
10005:{\displaystyle c(E)=c(E')\smile c(E'')}
9212:coefficients. One can also define the
9197:{\displaystyle H^{2k}(X;\mathbb {Z} ),}
7267:
5509:are the first Chern classes. The roots
4357:{\displaystyle c=1+c_{1}+c_{2}+\cdots }
4306:By the additivity of total Chern class
4041:where η is the vector bundle such that
2288:. We prove the complex version of the
265:For a sheaf theoretic description, see
16219:
15732:A Concise Course in Algebraic Topology
15678:
14787:
14716:
13224:
12508:can be represented by five points (by
11626:For example, consider the nonsingular
10755:
10523:in the sense of Grothendieck, denoted
10497:{\displaystyle H^{*}(\mathbb {P} (E))}
7081:{\displaystyle H^{*}(-,\mathbb {Z} ).}
6254:, whose first few terms are: (we drop
4728:{\displaystyle c_{1}(E^{*})=-c_{1}(E)}
878:To say that the expression given is a
287:An important special case occurs when
176:It can be shown that for any two maps
15821:
15801:â A downloadable book-in-progress by
15773:
15571:
15081:
14774:The sequence is sometimes called the
14500:
11699:and we have the short exact sequence
11692:{\displaystyle {\mathcal {O}}_{X}(5)}
11659:. Then the normal bundle is given by
11353:{\displaystyle i>{\text{rank}}(E)}
11059:{\displaystyle {\mathcal {O}}_{X}(D)}
10877:of the underlying real vector bundle.
10208:is the projective space of the fiber
10124:of the underlying real vector bundle.
7370:
5502:{\displaystyle a_{i}(E)=c_{1}(L_{i})}
3268:be the canonical projection, and let
2273:{\displaystyle a\partial /\partial z}
1444:
905:If follows from the matrix identity
570:are given as the coefficients of the
355:{\displaystyle H^{2}(X;\mathbb {Z} )}
77:
71:
15813:Chern numbers of algebraic varieties
15699:
14627:if and only if the Chern numbers of
13219:
11621:
10972:{\displaystyle c_{i}(E)\in A^{i}(X)}
5032:simply keeps track of the degree of
2239:{\displaystyle V=T\mathbb {CP} ^{1}}
706:The determinant is over the ring of
96:The converse, however, is not true.
15725:
14744:
14239:The Chern character is used in the
13478:{\displaystyle x_{i}=c_{1}(L_{i}),}
10342:{\displaystyle \mathbb {P} (E_{b})}
10113:{\displaystyle e(E_{\mathbb {R} })}
4893:
3188:{\displaystyle \mathbb {C} ^{n+1},}
3150:{\displaystyle z_{0},\ldots ,z_{n}}
3099:) and the last nonzero term is the
66:. Chern classes were introduced by
13:
15679:Hartshorne, Robin (29 June 2013).
15612:
15005:
14993:
14981:
14969:
14849:
14220:) into the rational cohomology of
13997:
13572:
13314:
12849:
12793:
12747:
12718:
12666:
12628:
12611:
12413:
12346:
12059:
11903:
11880:
11770:
11732:
11715:
11669:
11575:
11537:
11520:
11504:there is the short exact sequence
11464:
11113:
11036:
9124:are a sequence of elements of the
9065:
8968:
8922:
8893:
7805:
4845:
4620:
4454:
4409:
4406:
4403:
4253:
4228:
4206:
4166:
4147:
4079:
4050:
4005:
3894:{\displaystyle \mathbb {C} ^{n+1}}
3788:
3724:
3631:
3559:{\displaystyle {\mathcal {O}}_{U}}
3545:
3490:
3049:
3006:
2905:
2876:
2845:{\displaystyle T\mathbb {CP} ^{1}}
2656:
2515:
2264:
2256:
1822:
1758:
1589:and define the new vector bundle:
1537:be a complex vector bundle over a
1377:
1370:
1367:
1357:
1350:
1347:
1334:
1326:
1323:
1304:
1296:
1293:
1236:
1229:
1226:
1210:
1202:
1199:
1172:
1165:
1162:
1013:
1010:
916:
913:
860:the sum from the determinant, and
774:
750:
627:
584:
276:be a topological space having the
14:
16243:
15792:
15639:"La théorie des classes de Chern"
14646:have a well-defined Chern class.
14575:is a complex vector bundle. The
10215:. The total space of this bundle
9783:{\displaystyle \mathbb {CP} ^{k}}
8877:{\displaystyle \mathbb {CP} ^{1}}
6982:is a natural transformation from
6250:is called the Chern character of
4364:(i.e., the Whitney sum formula),
3296:
3226:
2179:{\displaystyle \mathbb {CP} ^{1}}
1573:
1530:{\displaystyle \pi \colon E\to B}
226:Chern classes arise naturally in
16232:Chinese mathematical discoveries
16200:
16173:
16163:
16153:
16142:
16132:
16131:
15925:
15094:elementary symmetric polynomials
14638:The theory also extends to real
14246:
12032:{\displaystyle \mathbb {P} ^{4}}
11652:{\displaystyle \mathbb {P} ^{4}}
11273:{\displaystyle c(E)=c(E')c(E'')}
10584:by expanding this way the class
10304:{\displaystyle c_{1}(\tau )=:-a}
6711:{\displaystyle f_{E}:X\to G_{n}}
5663:elementary symmetric polynomials
5081:is completely determined by the
2852:is not a trivial vector bundle.
2400:{\displaystyle c_{1}(V)\not =0.}
1456:The basic observation is that a
486:
15735:, University of Chicago Press,
15210:In particular, on the one hand
15074:
15011:
14999:
14956:is homogeneous and has degree |
14479:Generalized cohomology theories
14241:HirzebruchâRiemannâRoch theorem
11857:{\displaystyle A^{\bullet }(X)}
11828:denote the hyperplane class in
11214:the Whitney sum formula holds:
10239:{\displaystyle \mathbb {P} (E)}
10159:{\displaystyle \mathbb {P} (E)}
9866:Grothendieck axiomatic approach
3617:, fits into the exact sequence
1682:has rank one less than that of
1582:{\displaystyle B'=E\setminus B}
261:The Chern class of line bundles
121:
15529:
15497:
15453:
15434:
15400:
15387:
15350:
15318:
15305:
15299:
15269:
15263:
15226:
15220:
15194:
15162:
15146:
15140:
14987:
14975:
14963:
14928:
14920:
14908:
14899:
14893:
14890:
14887:
14873:
14857:
14836:
14833:
14819:
14793:
14781:
14768:
14759:
14738:
14710:
14179:
14173:
14164:
14158:
14146:
14134:
14109:
14103:
14091:
14085:
14073:
14061:
13959:
13953:
13897:
13894:
13888:
13869:
13863:
13850:
13844:
13819:
13812:
13799:
13783:
13780:
13774:
13749:
13742:
13729:
13713:
13707:
13691:
13685:
13673:
13667:
13634:
13592:
13504:
13498:
13469:
13456:
13339:
13332:
13292:
13289:
13283:
13270:
13258:
13252:
13120:
13113:
13110:
13082:
13076:
13070:
13067:
13055:
13036:
13027:
13020:
13004:
12998:
12986:
12983:
12974:
12967:
12958:
12952:
12940:
12924:
12921:
12915:
12903:
12892:
12888:
12882:
12873:
12860:
12843:
12829:we can then calculate this as
12813:
12810:
12804:
12787:
12779:
12767:
12741:
12729:
12712:
12686:
12683:
12677:
12660:
12650:
12622:
12605:
12441:
12435:
12424:
12407:
12392:
12386:
12357:
12340:
12070:
12053:
11942:
11929:
11923:
11920:
11914:
11897:
11891:
11874:
11851:
11845:
11790:
11787:
11781:
11764:
11754:
11726:
11709:
11686:
11680:
11603:
11569:
11559:
11531:
11514:
11448:
11442:
11429:
11426:
11420:
11384:
11378:
11372:
11347:
11341:
11304:
11298:
11267:
11256:
11250:
11239:
11230:
11224:
11198:
11187:
11181:
11170:
11145:
11139:
11133:
11130:
11124:
11107:
11053:
11047:
11005:
10999:
10966:
10960:
10944:
10938:
10905:
10852:
10846:
10795:
10789:
10734:
10728:
10706:
10700:
10653:
10647:
10571:
10565:
10546:
10540:
10491:
10488:
10482:
10474:
10457:then states that any class in
10440:
10437:
10431:
10423:
10336:
10323:
10289:
10283:
10233:
10227:
10153:
10147:
10107:
10092:
10069:
10054:
10039:
10033:
9999:
9988:
9979:
9968:
9959:
9953:
9923:
9912:
9906:
9895:
9736:
9730:
9708:
9702:
9665:
9653:
9620:
9614:
9605:
9599:
9590:
9578:
9523:
9494:
9488:
9462:
9446:
9397:
9355:
9349:
9325:Classical axiomatic definition
9297:
9291:
9275:
9269:
9253:
9247:
9231:
9225:
9188:
9174:
9079:
9076:
9070:
9060:
9030:
9004:
8982:
8979:
8973:
8963:
8933:
8927:
8908:
8898:
8695:
8676:
8537:
8518:
8472:{\displaystyle c_{i}=c_{i}(E)}
8466:
8460:
8412:
8406:
8390:
8374:
8337:
8331:
8285:
8276:
8230:
8214:
8194:
8188:
8142:
8133:
8087:
8068:
7957:
7950:
7937:
7931:
7876:
7869:
7856:
7850:
7765:
7759:
7740:
7734:
7674:
7662:
7610:
7603:
7590:
7584:
7547:
7535:
7498:
7492:
7473:
7463:
7457:
7444:
7346:
7340:
7303:
7297:
7249:
7217:
7206:
7185:
7169:
7166:
7152:
7136:
7117:
7114:
7072:
7058:
7028:
7009:
6911:
6898:
6877:
6871:
6838:
6824:
6811:
6808:
6776:
6695:
6612:
6581:
6476:
6470:
6422:
6362:
6346:
6312:
6277:
6271:
6226:
6223:
6217:
6195:
6189:
6176:
6155:
6149:
6120:
6114:
6093:
6087:
5970:
5967:
5935:
5913:
5881:
5868:
5738:
5732:
5665:. In other words, thinking of
5642:
5639:
5633:
5611:
5605:
5592:
5576:
5570:
5532:
5526:
5496:
5483:
5467:
5461:
5431:
5425:
5419:
5400:
5394:
5388:
5382:
5363:
5357:
5351:
5266:
5255:
5242:
5236:
5220:
5203:
5161:
5155:
5133:
5127:
5105:
5099:
5068:
5062:
5000:
4994:
4969:
4963:
4941:
4935:
4832:
4812:
4722:
4716:
4697:
4684:
4654:
4645:
4597:
4569:
4520:
4507:
4489:
4485:
4479:
4448:
4439:
4418:
4394:
4374:
4284:
4272:
4265:
4258:
4245:
4242:
4233:
4220:
4211:
4201:
4189:
4180:
4171:
4161:
4100:
4088:
4064:
4055:
4028:
4019:
4010:
4000:
3799:
3769:
3755:
3743:
3738:
3729:
3685:
3668:
3656:
3627:
3610:{\displaystyle d(z_{i}/z_{0})}
3604:
3576:
3515:
3380:
3352:
3238:
3078:
3072:
2980:
2959:
2954:
2942:
2935:
2928:
2899:
2870:
2774:
2763:
2754:
2744:
2736:
2583:
2572:
2563:
2553:
2545:
2481:
2470:
2461:
2451:
2443:
2388:
2382:
2344:
2318:
2083:
2068:
2047:
2036:
1996:
1975:
1969:
1910:{\displaystyle \pi |_{B'}^{*}}
1887:
1856:
1853:
1834:
1798:
1789:
1784:
1770:
1754:
1728:
1702:
1604:
1521:
1488:
1480:
1451:orientation of a vector bundle
1381:
1374:
1360:
1354:
1343:
1330:
1313:
1300:
1240:
1233:
1219:
1206:
1175:
1169:
1128:
1122:
1080:
1068:
1038:
1035:
1032:
1026:
1017:
1006:
994:
988:
965:
962:
956:
950:
938:
935:
929:
920:
811:
799:
681:
675:
553:
547:
445:
434:
418:
412:
396:
379:
349:
335:
1:
15799:Vector Bundles & K-Theory
15565:
15558:, giving the claimed formula.
14750:"Vector Bundles and K-theory"
14485:generalized cohomology theory
13929:for arbitrary vector bundles
13925:, is taken as the definition
11390:{\displaystyle E\mapsto c(E)}
9101:
6672:, there is a continuous map
890:. That is, Chern classes are
219:There is also an approach of
161:is equal to the pullback, by
15847:
14236:is a homomorphism of rings.
10266:, and the first Chern class
9882:Naturality: (Same as above)
6722:says the cohomology ring of
2670:{\displaystyle c_{1}=\left.}
2205:for the Riemann sphere. Let
7:
15613:Fulton, W. (29 June 2013).
15064:{\displaystyle \alpha _{i}}
14661:
11397:extends to a ring morphism
9143:, which is usually denoted
7276:be a vector bundle of rank
5762:at a complex vector bundle
3984:can be identified with the
2144:
109:AtiyahâSinger index theorem
10:
16248:
16094:Banach fixed-point theorem
14535:is a homomorphism from CH(
14391:{\displaystyle c_{1}c_{2}}
14323:orientation homology class
13652:This can be rewritten as:
11316:{\displaystyle c_{i}(E)=0}
11017:{\displaystyle c_{0}(E)=1}
10807:{\displaystyle c_{k}(V)=0}
9872:Alexander Grothendieck
9367:{\displaystyle c_{0}(E)=1}
7041:to the cohomology functor
5186:is additive in the sense:
495:
267:Exponential sheaf sequence
264:
16127:
16084:
16048:
15934:
15923:
15855:
14631:âČ coincide with those of
14354:{\displaystyle c_{1}^{3}}
14321:) can be paired with the
10873:) is always equal to the
10351:complex projective spaces
10182:whose fiber at any point
9561:{\displaystyle E\oplus F}
1945:{\displaystyle k<2n-1}
1552:as the zero section, let
725:{\displaystyle n\times n}
572:characteristic polynomial
492:Via the ChernâWeil theory
83:Basic idea and motivation
15285:while on the other hand
14703:
14562:almost complex manifolds
14552:Manifolds with structure
13163:is a spin 4-manifold if
11026:For an invertible sheaf
10858:{\displaystyle c_{n}(V)}
10311:restricts on each fiber
9943:of vector bundles, then
9756:tautological line bundle
9406:{\displaystyle f:Y\to X}
8848:Applications of formulae
5744:{\displaystyle c_{t}(E)}
5538:{\displaystyle a_{i}(E)}
5074:{\displaystyle c_{t}(E)}
2856:Complex projective space
2192:complex projective space
1653:and a nonzero vector on
1645:is specified by a fiber
1618:{\displaystyle E'\to B'}
1086:{\displaystyle \ln(X+I)}
852:is used here only as an
559:{\displaystyle c_{k}(V)}
470:holomorphic line bundles
64:GromovâWitten invariants
15635:Grothendieck, Alexander
15082:§ Chern polynomial
14790:, Ch. II. Theorem 8.13.
14611:, giving an integer, a
10825:The top Chern class of
10768:is the complex rank of
10178:as the fiber bundle on
9120:, the Chern classes of
7518:is a line bundle, then
6718:unique up to homotopy.
4788:{\displaystyle k\geq 0}
4769:In particular, for any
888:exact differential form
756:{\displaystyle \Omega }
590:{\displaystyle \Omega }
16227:Characteristic classes
16149:Mathematics portal
16049:Metrics and properties
16035:Second-countable space
15758:Characteristic classes
15540:
15486:
15433:
15383:
15279:
15252:
15204:
15117:
15065:
15038:
14946:
14466:
14442:
14419:
14392:
14355:
14315:
14295:
14272:
14189:
14116:
14029:
13913:
13644:
13576:
13479:
13420:
13366:
13318:
13206:
13183:
13157:
13137:
12823:
12696:
12589:
12569:
12536:Degree d hypersurfaces
12526:
12502:
12475:
12364:
12315:
12033:
12002:
11858:
11822:
11800:
11693:
11653:
11613:
11498:
11455:
11391:
11354:
11317:
11274:
11208:
11152:
11080:
11060:
11018:
10973:
10915:
10914:{\displaystyle E\to X}
10859:
10808:
10744:
10608:
10607:{\displaystyle -a^{n}}
10578:
10498:
10447:
10343:
10305:
10260:
10240:
10202:
10201:{\displaystyle b\in B}
10170:complex vector bundle
10160:
10114:
10076:
10006:
9933:
9855:
9784:
9746:
9691:
9630:
9562:
9533:
9532:{\displaystyle F\to X}
9501:
9423:vector bundle pullback
9407:
9368:
9313:
9198:
9092:
9037:
8940:
8878:
8827:
8660:
8632:
8502:
8473:
8422:
8351:
8041:
8021:
7973:
7914:
7797:
7711:
7639:
7573:
7505:
7421:
7401:
7363:
7329:
7259:
7082:
7035:
6960:
6921:
6848:
6737:; so, the pullback of
6712:
6628:
6496:
6438:
6244:
6064:
5977:
5801:: We have polynomials
5745:
5649:
5539:
5503:
5438:
5328:
5276:
5168:
5075:
5020:
4885:
4789:
4756:
4729:
4661:
4604:
4542:
4358:
4296:
4109:
4035:
3978:
3933:
3895:
3853:
3825:
3611:
3560:
3529:
3473:
3325:
3262:
3189:
3157:be the coordinates of
3151:
3093:Serre's twisting sheaf
3085:
3033:
2990:
2846:
2796:
2671:
2602:
2500:
2401:
2357:
2274:
2240:
2180:
2126:
1946:
1917:is an isomorphism for
1911:
1869:
1738:
1676:
1619:
1583:
1531:
1495:
1436:
1087:
1045:
972:
818:
757:
726:
698:
591:
560:
455:
356:
221:Alexander Grothendieck
93:topological invariants
89:characteristic classes
41:characteristic classes
15778:, Walter De Gruyter,
15774:Rubei, Elena (2014),
15578:Annals of Mathematics
15541:
15466:
15413:
15363:
15280:
15232:
15205:
15118:
15116:{\displaystyle x_{i}}
15084:.) Observe that when
15066:
15039:
15037:{\displaystyle c_{i}}
14947:
14720:; Tu, Loring (1995).
14698:Localized Chern class
14673:StiefelâWhitney class
14467:
14443:
14420:
14418:{\displaystyle c_{3}}
14393:
14356:
14316:
14296:
14273:
14190:
14117:
14030:
13914:
13645:
13556:
13480:
13421:
13367:
13298:
13207:
13184:
13158:
13138:
12824:
12697:
12590:
12570:
12527:
12503:
12501:{\displaystyle h^{3}}
12476:
12365:
12316:
12039:. This gives us that
12034:
12003:
11859:
11823:
11801:
11694:
11654:
11614:
11499:
11456:
11392:
11355:
11318:
11275:
11209:
11153:
11081:
11061:
11019:
10974:
10916:
10887:Axiomatic description
10882:In algebraic geometry
10860:
10809:
10745:
10614:, with the relation:
10609:
10579:
10499:
10448:
10344:
10306:
10261:
10259:{\displaystyle \tau }
10241:
10203:
10161:
10115:
10077:
10007:
9934:
9856:
9785:
9747:
9671:
9631:
9563:
9534:
9502:
9408:
9369:
9314:
9199:
9108:complex vector bundle
9093:
9038:
8941:
8879:
8841:Segre class#Example 2
8828:
8661:
8633:
8503:
8474:
8423:
8352:
8042:
8022:
7974:
7894:
7777:
7712:
7640:
7553:
7506:
7422:
7402:
7400:{\displaystyle E^{*}}
7364:
7309:
7260:
7083:
7036:
6961:
6922:
6849:
6713:
6650:infinite Grassmannian
6629:
6482:
6439:
6245:
6065:
5978:
5746:
5691:symmetric polynomials
5674:as formal variables,
5650:
5540:
5504:
5439:
5329:
5277:
5169:
5076:
5021:
4886:
4790:
4757:
4755:{\displaystyle E^{*}}
4730:
4662:
4605:
4543:
4359:
4297:
4110:
4036:
3979:
3934:
3896:
3854:
3852:{\displaystyle e_{i}}
3826:
3612:
3561:
3530:
3474:
3326:
3263:
3190:
3152:
3086:
3034:
2991:
2847:
2797:
2672:
2603:
2501:
2402:
2358:
2280:at each point, where
2275:
2241:
2181:
2127:
1947:
1912:
1870:
1739:
1690:for the fiber bundle
1677:
1620:
1584:
1548:as being embedded in
1532:
1496:
1458:complex vector bundle
1437:
1088:
1046:
973:
819:
758:
727:
699:
592:
561:
456:
357:
257:for more discussion.
149:, there exists a map
68:Shiing-Shen Chern
29:differential geometry
16104:Invariance of domain
16056:Euler characteristic
16030:Bundle (mathematics)
15750:Milnor, John Willard
15289:
15214:
15127:
15100:
15092:can be expressed as
15048:
15021:
14803:
14644:symplectic manifolds
14456:
14429:
14402:
14365:
14333:
14305:
14282:
14259:
14125:
14052:
13944:
13658:
13489:
13430:
13378:
13243:
13231:topological K-theory
13193:
13167:
13147:
12833:
12706:
12702:giving the relation
12599:
12579:
12544:
12516:
12485:
12378:
12327:
12043:
12014:
11868:
11832:
11812:
11703:
11663:
11634:
11508:
11473:
11401:
11366:
11327:
11285:
11218:
11164:
11094:
11070:
11030:
10986:
10925:
10899:
10833:
10776:
10618:
10588:
10527:
10461:
10455:LerayâHirsch theorem
10360:
10315:
10270:
10250:
10219:
10186:
10139:
10130:LerayâHirsch theorem
10086:
10027:
9947:
9889:
9809:
9762:
9640:
9572:
9546:
9517:
9433:
9385:
9336:
9219:
9158:
9154:), is an element of
9047:
8950:
8888:
8856:
8670:
8644:
8512:
8486:
8434:
8361:
8051:
8031:
7985:
7981:For the Chern roots
7721:
7649:
7522:
7431:
7411:
7384:
7284:
7268:Computation formulae
7105:
7045:
6986:
6980:characteristic class
6936:
6858:
6750:
6676:
6461:
6262:
6078:
5987:
5813:
5776:and then replacing Ï
5770:as a polynomial in Ï
5719:
5557:
5513:
5448:
5338:
5286:
5190:
5093:
5049:
4922:
4799:
4773:
4739:
4671:
4614:
4556:
4368:
4310:
4119:
4045:
3991:
3951:
3909:
3870:
3836:
3621:
3570:
3539:
3486:
3479:In other words, the
3336:
3272:
3199:
3161:
3115:
3043:
3000:
2864:
2821:
2683:
2614:
2512:
2416:
2369:
2305:
2250:
2209:
2158:
2139:The Thom isomorphism
1956:
1921:
1879:
1748:
1694:
1661:
1593:
1556:
1509:
1464:
1099:
1059:
1051:. Now applying the
982:
909:
771:
747:
710:
607:
581:
534:
366:
322:
318:and the elements of
232:locally free sheaves
105:RiemannâRoch theorem
101:linearly independent
16114:Tychonoff's theorem
16109:Poincaré conjecture
15863:General (point-set)
15616:Intersection Theory
15548:Newton's identities
15008:, Remark 3.2.3. (c)
14984:, Remark 3.2.3. (b)
14972:, Remark 3.2.3. (a)
14853:
14755:. Proposition 3.10.
14693:Quantum Hall effect
14350:
14043:of the connection.
13923:splitting principle
13633:
13609:
13374:More generally, if
13225:The Chern character
13182:{\displaystyle 4-d}
12481:since the class of
10756:The top Chern class
10128:He shows using the
10015:Normalization: If
8777:
8740:
8659:{\displaystyle r=3}
8582:
8501:{\displaystyle r=2}
7005:
6955:
6897:
6767:
6598:
6379:
6329:
6072:Newton's identities
6043:
5854:
5830:
5713:splitting principle
3903:elementary geometry
3721:
3566:-module with basis
3453:
2802:after switching to
2015:
1906:
1817:
834:exterior derivative
255:ChernâSimons theory
56:ChernâSimons theory
23:, in particular in
16099:De Rham cohomology
16020:Polyhedral complex
16010:Simplicial complex
15754:Stasheff, James D.
15682:Algebraic Geometry
15656:10.24033/bsmf.1501
15573:Chern, Shiing-Shen
15536:
15534:
15275:
15200:
15113:
15061:
15034:
14942:
14839:
14591:and of dimension 2
14501:Algebraic geometry
14490:complex orientable
14462:
14441:{\displaystyle 2n}
14438:
14415:
14388:
14351:
14336:
14311:
14294:{\displaystyle 2n}
14291:
14271:{\displaystyle 2n}
14268:
14185:
14112:
14025:
13909:
13640:
13619:
13595:
13475:
13416:
13362:
13205:{\displaystyle 2k}
13202:
13179:
13153:
13133:
13131:
12819:
12692:
12585:
12565:
12522:
12498:
12471:
12360:
12311:
12309:
12029:
11998:
11854:
11818:
11796:
11689:
11649:
11609:
11494:
11451:
11387:
11350:
11313:
11270:
11204:
11148:
11076:
11056:
11014:
10969:
10911:
10855:
10804:
10740:
10604:
10574:
10494:
10443:
10339:
10301:
10256:
10236:
10198:
10156:
10110:
10072:
10002:
9929:
9851:
9780:
9742:
9626:
9558:
9529:
9497:
9403:
9364:
9309:
9204:the cohomology of
9194:
9088:
9033:
8936:
8874:
8823:
8763:
8726:
8656:
8628:
8568:
8498:
8469:
8418:
8347:
8345:
8275:
8132:
8037:
8017:
7969:
7707:
7635:
7501:
7417:
7397:
7359:
7255:
7078:
7031:
6989:
6956:
6939:
6917:
6883:
6844:
6753:
6708:
6624:
6584:
6434:
6365:
6315:
6240:
6060:
6029:
5973:
5840:
5816:
5741:
5689:. A basic fact on
5645:
5535:
5499:
5434:
5324:
5272:
5164:
5071:
5045:). In particular,
5016:
4881:
4785:
4752:
4725:
4657:
4600:
4538:
4354:
4292:
4105:
4031:
3974:
3929:
3891:
3849:
3821:
3701:
3607:
3556:
3535:, which is a free
3525:
3469:
3439:
3321:
3258:
3185:
3147:
3081:
3029:
2986:
2842:
2792:
2667:
2598:
2496:
2397:
2353:
2291:hairy ball theorem
2270:
2236:
2176:
2122:
2117:
1994:
1942:
1907:
1885:
1865:
1796:
1734:
1675:{\displaystyle E'}
1672:
1615:
1579:
1527:
1491:
1445:Via an Euler class
1432:
1111:
1083:
1041:
968:
896:de Rham cohomology
892:cohomology classes
814:
753:
722:
694:
664:
587:
556:
474:linear equivalence
451:
352:
308:complete invariant
243:hairy ball theorem
228:algebraic geometry
208:are well-defined.
87:Chern classes are
78:Geometric approach
33:algebraic geometry
25:algebraic topology
16214:
16213:
16003:fundamental group
15785:978-3-11-031622-3
15767:978-0-691-08122-9
15718:978-3-540-25907-7
15692:978-1-4757-3849-0
15626:978-3-662-02421-8
15581:, Second Series,
14688:Schubert calculus
14656:Arakelov geometry
14605:fundamental class
14599:of total degree 2
14518:l-adic cohomology
14465:{\displaystyle n}
14314:{\displaystyle n}
14253:oriented manifold
14251:If we work on an
14009:
13938:ChernâWeil theory
13797:
13727:
13590:
13357:
13220:Proximate notions
13156:{\displaystyle X}
12928:
12817:
12588:{\displaystyle d}
12525:{\displaystyle X}
12142:
11821:{\displaystyle h}
11628:quintic threefold
11622:Quintic threefold
11339:
11079:{\displaystyle D}
9214:total Chern class
9115:topological space
9098:for any integer.
8430:For example, for
8240:
8097:
8040:{\displaystyle E}
7826:
7420:{\displaystyle E}
6658:classifying space
6579:
6556:
6537:
6360:
6310:
5083:total Chern class
4860:
4413:
3783:
3699:
3454:
3097:hyperplane bundle
2817:This proves that
2804:polar coordinates
2784:
2739:
2710:
2648:
2593:
2548:
2491:
2446:
1819:
1539:paracompact space
1406:
1265:
1187:
1102:
797:
655:
639:
498:ChernâWeil theory
214:ChernâWeil theory
135:classifying space
16239:
16204:
16203:
16177:
16176:
16167:
16157:
16147:
16146:
16135:
16134:
15929:
15842:
15835:
15828:
15819:
15818:
15809:Dieter Kotschick
15788:
15770:
15745:
15721:
15707:(4th ed.),
15696:
15675:
15658:
15630:
15609:
15559:
15545:
15543:
15542:
15537:
15535:
15528:
15527:
15509:
15508:
15496:
15495:
15485:
15480:
15459:
15452:
15451:
15432:
15427:
15406:
15399:
15398:
15382:
15377:
15356:
15349:
15348:
15330:
15329:
15284:
15282:
15281:
15276:
15262:
15261:
15251:
15246:
15209:
15207:
15206:
15201:
15193:
15192:
15174:
15173:
15161:
15160:
15139:
15138:
15122:
15120:
15119:
15114:
15112:
15111:
15078:
15072:
15070:
15068:
15067:
15062:
15060:
15059:
15043:
15041:
15040:
15035:
15033:
15032:
15015:
15009:
15003:
14997:
14996:, Example 3.2.2.
14991:
14985:
14979:
14973:
14967:
14961:
14951:
14949:
14948:
14943:
14941:
14940:
14936:
14931:
14923:
14886:
14872:
14871:
14852:
14847:
14832:
14818:
14817:
14797:
14791:
14785:
14779:
14772:
14766:
14763:
14757:
14756:
14754:
14742:
14736:
14735:
14714:
14668:Pontryagin class
14514:etale cohomology
14495:formal group law
14471:
14469:
14468:
14463:
14447:
14445:
14444:
14439:
14424:
14422:
14421:
14416:
14414:
14413:
14397:
14395:
14394:
14389:
14387:
14386:
14377:
14376:
14360:
14358:
14357:
14352:
14349:
14344:
14320:
14318:
14317:
14312:
14300:
14298:
14297:
14292:
14277:
14275:
14274:
14269:
14194:
14192:
14191:
14186:
14121:
14119:
14118:
14113:
14038:
14034:
14032:
14031:
14026:
14024:
14020:
14019:
14015:
14014:
14010:
14008:
14000:
13992:
13918:
13916:
13915:
13910:
13887:
13886:
13862:
13861:
13843:
13842:
13827:
13826:
13811:
13810:
13798:
13790:
13773:
13772:
13757:
13756:
13741:
13740:
13728:
13720:
13706:
13705:
13649:
13647:
13646:
13641:
13632:
13627:
13608:
13603:
13591:
13589:
13578:
13575:
13570:
13552:
13551:
13550:
13549:
13526:
13525:
13524:
13523:
13484:
13482:
13481:
13476:
13468:
13467:
13455:
13454:
13442:
13441:
13425:
13423:
13422:
13417:
13415:
13414:
13396:
13395:
13371:
13369:
13368:
13363:
13358:
13356:
13348:
13347:
13346:
13331:
13330:
13320:
13317:
13312:
13282:
13281:
13211:
13209:
13208:
13203:
13188:
13186:
13185:
13180:
13162:
13160:
13159:
13154:
13142:
13140:
13139:
13134:
13132:
13128:
13127:
13109:
13108:
13042:
13035:
13034:
13019:
13018:
12982:
12981:
12933:
12929:
12927:
12901:
12900:
12899:
12871:
12859:
12858:
12853:
12852:
12828:
12826:
12825:
12820:
12818:
12816:
12803:
12802:
12797:
12796:
12782:
12778:
12777:
12776:
12775:
12770:
12764:
12763:
12758:
12751:
12750:
12736:
12728:
12727:
12722:
12721:
12701:
12699:
12698:
12693:
12676:
12675:
12670:
12669:
12659:
12658:
12653:
12647:
12646:
12645:
12644:
12639:
12632:
12631:
12621:
12620:
12615:
12614:
12594:
12592:
12591:
12586:
12574:
12572:
12571:
12566:
12564:
12563:
12558:
12531:
12529:
12528:
12523:
12510:BĂ©zout's theorem
12507:
12505:
12504:
12499:
12497:
12496:
12480:
12478:
12477:
12472:
12461:
12460:
12445:
12444:
12423:
12422:
12417:
12416:
12406:
12405:
12396:
12395:
12369:
12367:
12366:
12361:
12356:
12355:
12350:
12349:
12339:
12338:
12320:
12318:
12317:
12312:
12310:
12306:
12305:
12290:
12289:
12265:
12261:
12257:
12256:
12255:
12240:
12239:
12207:
12203:
12202:
12201:
12186:
12185:
12147:
12143:
12141:
12127:
12126:
12125:
12110:
12109:
12081:
12069:
12068:
12063:
12062:
12038:
12036:
12035:
12030:
12028:
12027:
12022:
12007:
12005:
12004:
11999:
11997:
11996:
11981:
11980:
11950:
11949:
11913:
11912:
11907:
11906:
11890:
11889:
11884:
11883:
11863:
11861:
11860:
11855:
11844:
11843:
11827:
11825:
11824:
11819:
11805:
11803:
11802:
11797:
11780:
11779:
11774:
11773:
11763:
11762:
11757:
11751:
11750:
11749:
11748:
11743:
11736:
11735:
11725:
11724:
11719:
11718:
11698:
11696:
11695:
11690:
11679:
11678:
11673:
11672:
11658:
11656:
11655:
11650:
11648:
11647:
11642:
11618:
11616:
11615:
11610:
11602:
11601:
11600:
11599:
11594:
11588:
11579:
11578:
11568:
11567:
11562:
11556:
11555:
11554:
11553:
11548:
11541:
11540:
11530:
11529:
11524:
11523:
11503:
11501:
11500:
11495:
11493:
11492:
11487:
11460:
11458:
11457:
11452:
11441:
11440:
11419:
11418:
11396:
11394:
11393:
11388:
11359:
11357:
11356:
11351:
11340:
11337:
11322:
11320:
11319:
11314:
11297:
11296:
11279:
11277:
11276:
11271:
11266:
11249:
11213:
11211:
11210:
11205:
11197:
11180:
11157:
11155:
11154:
11149:
11123:
11122:
11117:
11116:
11106:
11105:
11085:
11083:
11082:
11077:
11065:
11063:
11062:
11057:
11046:
11045:
11040:
11039:
11023:
11021:
11020:
11015:
10998:
10997:
10978:
10976:
10975:
10970:
10959:
10958:
10937:
10936:
10920:
10918:
10917:
10912:
10864:
10862:
10861:
10856:
10845:
10844:
10813:
10811:
10810:
10805:
10788:
10787:
10749:
10747:
10746:
10741:
10727:
10726:
10699:
10698:
10674:
10673:
10646:
10645:
10633:
10632:
10613:
10611:
10610:
10605:
10603:
10602:
10583:
10581:
10580:
10575:
10564:
10563:
10539:
10538:
10503:
10501:
10500:
10495:
10481:
10473:
10472:
10452:
10450:
10449:
10444:
10430:
10422:
10421:
10409:
10408:
10384:
10383:
10348:
10346:
10345:
10340:
10335:
10334:
10322:
10310:
10308:
10307:
10302:
10282:
10281:
10265:
10263:
10262:
10257:
10245:
10243:
10242:
10237:
10226:
10207:
10205:
10204:
10199:
10165:
10163:
10162:
10157:
10146:
10119:
10117:
10116:
10111:
10106:
10105:
10104:
10081:
10079:
10078:
10073:
10068:
10067:
10066:
10011:
10009:
10008:
10003:
9998:
9978:
9938:
9936:
9935:
9930:
9922:
9905:
9860:
9858:
9857:
9852:
9850:
9849:
9844:
9832:
9831:
9820:
9789:
9787:
9786:
9781:
9779:
9778:
9773:
9751:
9749:
9748:
9743:
9729:
9728:
9701:
9700:
9690:
9685:
9652:
9651:
9635:
9633:
9632:
9627:
9567:
9565:
9564:
9559:
9538:
9536:
9535:
9530:
9513:sum formula: If
9506:
9504:
9503:
9498:
9487:
9486:
9477:
9476:
9458:
9457:
9445:
9444:
9412:
9410:
9409:
9404:
9373:
9371:
9370:
9365:
9348:
9347:
9318:
9316:
9315:
9310:
9290:
9289:
9268:
9267:
9246:
9245:
9203:
9201:
9200:
9195:
9187:
9173:
9172:
9097:
9095:
9094:
9089:
9069:
9068:
9059:
9058:
9042:
9040:
9039:
9034:
9029:
9021:
9020:
9015:
9003:
9002:
8972:
8971:
8962:
8961:
8945:
8943:
8942:
8937:
8926:
8925:
8916:
8915:
8897:
8896:
8883:
8881:
8880:
8875:
8873:
8872:
8867:
8832:
8830:
8829:
8824:
8819:
8818:
8803:
8802:
8793:
8792:
8776:
8771:
8756:
8755:
8739:
8734:
8719:
8718:
8688:
8687:
8665:
8663:
8662:
8657:
8637:
8635:
8634:
8629:
8624:
8623:
8614:
8613:
8598:
8597:
8581:
8576:
8561:
8560:
8530:
8529:
8507:
8505:
8504:
8499:
8478:
8476:
8475:
8470:
8459:
8458:
8446:
8445:
8427:
8425:
8424:
8419:
8405:
8404:
8386:
8385:
8373:
8372:
8356:
8354:
8353:
8348:
8346:
8330:
8329:
8328:
8327:
8304:
8303:
8302:
8301:
8274:
8273:
8272:
8254:
8253:
8226:
8225:
8213:
8212:
8187:
8186:
8185:
8184:
8161:
8160:
8159:
8158:
8131:
8130:
8129:
8111:
8110:
8080:
8079:
8067:
8066:
8046:
8044:
8043:
8038:
8026:
8024:
8023:
8018:
8016:
8015:
7997:
7996:
7978:
7976:
7975:
7970:
7965:
7964:
7949:
7948:
7930:
7929:
7913:
7908:
7884:
7883:
7868:
7867:
7849:
7848:
7833:
7832:
7831:
7822:
7804:
7796:
7791:
7758:
7757:
7733:
7732:
7716:
7714:
7713:
7708:
7661:
7660:
7644:
7642:
7641:
7636:
7634:
7633:
7624:
7623:
7602:
7601:
7583:
7582:
7572:
7567:
7534:
7533:
7510:
7508:
7507:
7502:
7491:
7490:
7481:
7480:
7456:
7455:
7443:
7442:
7426:
7424:
7423:
7418:
7406:
7404:
7403:
7398:
7396:
7395:
7371:Chern polynomial
7368:
7366:
7365:
7360:
7358:
7357:
7339:
7338:
7328:
7323:
7296:
7295:
7264:
7262:
7261:
7256:
7248:
7247:
7229:
7228:
7216:
7205:
7197:
7196:
7184:
7183:
7165:
7151:
7150:
7135:
7134:
7087:
7085:
7084:
7079:
7071:
7057:
7056:
7040:
7038:
7037:
7032:
7027:
7026:
7004:
7003:
6997:
6965:
6963:
6962:
6957:
6954:
6953:
6947:
6926:
6924:
6923:
6918:
6910:
6909:
6896:
6891:
6870:
6869:
6853:
6851:
6850:
6845:
6837:
6823:
6822:
6807:
6806:
6788:
6787:
6775:
6766:
6761:
6717:
6715:
6714:
6709:
6707:
6706:
6688:
6687:
6633:
6631:
6630:
6625:
6611:
6610:
6597:
6592:
6580:
6572:
6567:
6566:
6557:
6549:
6538:
6536:
6535:
6534:
6533:
6532:
6508:
6507:
6498:
6495:
6490:
6443:
6441:
6440:
6435:
6421:
6420:
6405:
6404:
6395:
6394:
6378:
6373:
6361:
6353:
6345:
6344:
6328:
6323:
6311:
6303:
6298:
6297:
6249:
6247:
6246:
6241:
6233:
6216:
6215:
6188:
6187:
6175:
6174:
6159:
6158:
6148:
6147:
6124:
6123:
6113:
6112:
6069:
6067:
6066:
6061:
6059:
6058:
6042:
6037:
6025:
6024:
6012:
6011:
5999:
5998:
5982:
5980:
5979:
5974:
5966:
5965:
5947:
5946:
5934:
5933:
5912:
5911:
5893:
5892:
5880:
5879:
5867:
5866:
5853:
5848:
5829:
5824:
5750:
5748:
5747:
5742:
5731:
5730:
5654:
5652:
5651:
5646:
5632:
5631:
5604:
5603:
5591:
5590:
5569:
5568:
5544:
5542:
5541:
5536:
5525:
5524:
5508:
5506:
5505:
5500:
5495:
5494:
5482:
5481:
5460:
5459:
5443:
5441:
5440:
5435:
5418:
5417:
5381:
5380:
5350:
5349:
5333:
5331:
5330:
5325:
5323:
5322:
5304:
5303:
5281:
5279:
5278:
5273:
5265:
5254:
5253:
5235:
5234:
5219:
5202:
5201:
5174:and conversely.
5173:
5171:
5170:
5165:
5154:
5153:
5126:
5125:
5080:
5078:
5077:
5072:
5061:
5060:
5025:
5023:
5022:
5017:
5012:
5011:
4993:
4992:
4962:
4961:
4934:
4933:
4904:Chern polynomial
4894:Chern polynomial
4890:
4888:
4887:
4882:
4877:
4876:
4867:
4866:
4865:
4856:
4844:
4831:
4830:
4825:
4819:
4811:
4810:
4794:
4792:
4791:
4786:
4761:
4759:
4758:
4753:
4751:
4750:
4734:
4732:
4731:
4726:
4715:
4714:
4696:
4695:
4683:
4682:
4666:
4664:
4663:
4658:
4644:
4643:
4642:
4641:
4636:
4630:
4624:
4623:
4609:
4607:
4606:
4601:
4596:
4588:
4587:
4582:
4576:
4568:
4567:
4547:
4545:
4544:
4539:
4534:
4533:
4503:
4502:
4478:
4477:
4476:
4475:
4470:
4464:
4458:
4457:
4438:
4437:
4432:
4414:
4412:
4398:
4393:
4392:
4387:
4381:
4363:
4361:
4360:
4355:
4347:
4346:
4334:
4333:
4301:
4299:
4298:
4293:
4288:
4287:
4257:
4256:
4232:
4231:
4210:
4209:
4170:
4169:
4151:
4150:
4141:
4140:
4135:
4129:
4114:
4112:
4111:
4106:
4104:
4103:
4083:
4082:
4054:
4053:
4040:
4038:
4037:
4032:
4009:
4008:
3983:
3981:
3980:
3975:
3973:
3972:
3967:
3961:
3938:
3936:
3935:
3930:
3928:
3927:
3922:
3916:
3900:
3898:
3897:
3892:
3890:
3889:
3878:
3858:
3856:
3855:
3850:
3848:
3847:
3830:
3828:
3827:
3822:
3798:
3797:
3792:
3791:
3784:
3782:
3781:
3780:
3768:
3767:
3754:
3752:
3751:
3746:
3728:
3727:
3720:
3709:
3700:
3698:
3697:
3696:
3684:
3683:
3667:
3665:
3664:
3659:
3653:
3652:
3651:
3650:
3645:
3639:
3616:
3614:
3613:
3608:
3603:
3602:
3593:
3588:
3587:
3565:
3563:
3562:
3557:
3555:
3554:
3549:
3548:
3534:
3532:
3531:
3526:
3524:
3523:
3518:
3512:
3511:
3510:
3509:
3504:
3498:
3478:
3476:
3475:
3470:
3455:
3452:
3447:
3438:
3437:
3436:
3424:
3423:
3411:
3410:
3398:
3397:
3387:
3379:
3378:
3369:
3364:
3363:
3348:
3347:
3331:. Then we have:
3330:
3328:
3327:
3322:
3311:
3310:
3295:
3294:
3289:
3267:
3265:
3264:
3259:
3257:
3256:
3251:
3245:
3225:
3224:
3213:
3194:
3192:
3191:
3186:
3181:
3180:
3169:
3156:
3154:
3153:
3148:
3146:
3145:
3127:
3126:
3090:
3088:
3087:
3082:
3071:
3070:
3069:
3068:
3063:
3053:
3052:
3038:
3036:
3035:
3030:
3028:
3027:
3026:
3025:
3020:
3010:
3009:
2995:
2993:
2992:
2987:
2979:
2978:
2973:
2958:
2957:
2927:
2926:
2925:
2924:
2919:
2909:
2908:
2898:
2897:
2896:
2895:
2890:
2880:
2879:
2851:
2849:
2848:
2843:
2841:
2840:
2835:
2801:
2799:
2798:
2793:
2785:
2783:
2782:
2781:
2772:
2771:
2766:
2757:
2742:
2741:
2740:
2732:
2716:
2711:
2703:
2698:
2697:
2676:
2674:
2673:
2668:
2663:
2659:
2649:
2647:
2636:
2626:
2625:
2607:
2605:
2604:
2599:
2594:
2592:
2591:
2590:
2581:
2580:
2575:
2566:
2551:
2550:
2549:
2541:
2522:
2505:
2503:
2502:
2497:
2492:
2490:
2489:
2488:
2479:
2478:
2473:
2464:
2449:
2448:
2447:
2439:
2426:
2406:
2404:
2403:
2398:
2381:
2380:
2362:
2360:
2359:
2354:
2343:
2335:
2334:
2329:
2317:
2316:
2279:
2277:
2276:
2271:
2263:
2245:
2243:
2242:
2237:
2235:
2234:
2229:
2203:local coordinate
2190:: 1-dimensional
2185:
2183:
2182:
2177:
2175:
2174:
2169:
2131:
2129:
2128:
2123:
2121:
2120:
2082:
2081:
2080:
2046:
2035:
2034:
2025:
2024:
2016:
2014:
2009:
2008:
1999:
1968:
1967:
1951:
1949:
1948:
1943:
1916:
1914:
1913:
1908:
1905:
1900:
1899:
1890:
1874:
1872:
1871:
1866:
1852:
1844:
1830:
1829:
1820:
1818:
1816:
1811:
1810:
1801:
1788:
1783:
1766:
1765:
1743:
1741:
1740:
1735:
1727:
1716:
1715:
1714:
1705:
1681:
1679:
1678:
1673:
1671:
1624:
1622:
1621:
1616:
1614:
1603:
1588:
1586:
1585:
1580:
1566:
1536:
1534:
1533:
1528:
1500:
1498:
1497:
1492:
1487:
1476:
1475:
1441:
1439:
1438:
1433:
1428:
1424:
1417:
1416:
1407:
1405:
1404:
1403:
1390:
1389:
1388:
1373:
1353:
1342:
1341:
1329:
1312:
1311:
1299:
1284:
1276:
1275:
1266:
1264:
1263:
1262:
1249:
1248:
1247:
1232:
1218:
1217:
1205:
1196:
1188:
1186:
1178:
1168:
1159:
1140:
1139:
1121:
1120:
1110:
1092:
1090:
1089:
1084:
1053:Maclaurin series
1050:
1048:
1047:
1042:
1016:
977:
975:
974:
969:
919:
894:in the sense of
823:
821:
820:
815:
798:
790:
762:
760:
759:
754:
731:
729:
728:
723:
703:
701:
700:
695:
693:
692:
674:
673:
663:
651:
647:
640:
638:
630:
619:
596:
594:
593:
588:
565:
563:
562:
557:
546:
545:
502:Given a complex
460:
458:
457:
452:
444:
433:
432:
411:
410:
395:
378:
377:
361:
359:
358:
353:
348:
334:
333:
145:over a manifold
43:associated with
16247:
16246:
16242:
16241:
16240:
16238:
16237:
16236:
16217:
16216:
16215:
16210:
16141:
16123:
16119:Urysohn's lemma
16080:
16044:
15930:
15921:
15893:low-dimensional
15851:
15846:
15795:
15786:
15768:
15743:
15719:
15709:Springer-Verlag
15693:
15627:
15591:10.2307/1969037
15568:
15563:
15562:
15533:
15532:
15523:
15519:
15504:
15500:
15491:
15487:
15481:
15470:
15457:
15456:
15447:
15443:
15428:
15417:
15404:
15403:
15394:
15390:
15378:
15367:
15354:
15353:
15344:
15340:
15325:
15321:
15308:
15292:
15290:
15287:
15286:
15257:
15253:
15247:
15236:
15215:
15212:
15211:
15188:
15184:
15169:
15165:
15156:
15152:
15134:
15130:
15128:
15125:
15124:
15107:
15103:
15101:
15098:
15097:
15079:
15075:
15055:
15051:
15049:
15046:
15045:
15028:
15024:
15022:
15019:
15018:
15016:
15012:
15004:
15000:
14992:
14988:
14980:
14976:
14968:
14964:
14932:
14927:
14919:
14918:
14914:
14882:
14864:
14860:
14848:
14843:
14828:
14810:
14806:
14804:
14801:
14800:
14798:
14794:
14786:
14782:
14773:
14769:
14764:
14760:
14752:
14743:
14739:
14732:
14715:
14711:
14706:
14664:
14652:
14560:invariants for
14554:
14503:
14481:
14457:
14454:
14453:
14430:
14427:
14426:
14409:
14405:
14403:
14400:
14399:
14382:
14378:
14372:
14368:
14366:
14363:
14362:
14345:
14340:
14334:
14331:
14330:
14306:
14303:
14302:
14283:
14280:
14279:
14260:
14257:
14256:
14249:
14126:
14123:
14122:
14053:
14050:
14049:
14036:
14001:
13993:
13991:
13987:
13980:
13976:
13969:
13965:
13945:
13942:
13941:
13882:
13878:
13857:
13853:
13838:
13834:
13822:
13818:
13806:
13802:
13789:
13768:
13764:
13752:
13748:
13736:
13732:
13719:
13701:
13697:
13659:
13656:
13655:
13628:
13623:
13604:
13599:
13582:
13577:
13571:
13560:
13545:
13541:
13540:
13536:
13519:
13515:
13514:
13510:
13490:
13487:
13486:
13463:
13459:
13450:
13446:
13437:
13433:
13431:
13428:
13427:
13410:
13406:
13391:
13387:
13379:
13376:
13375:
13349:
13342:
13338:
13326:
13322:
13321:
13319:
13313:
13302:
13277:
13273:
13244:
13241:
13240:
13227:
13222:
13194:
13191:
13190:
13168:
13165:
13164:
13148:
13145:
13144:
13130:
13129:
13123:
13119:
13104:
13100:
13040:
13039:
13030:
13026:
13014:
13010:
12977:
12973:
12931:
12930:
12902:
12895:
12891:
12872:
12870:
12863:
12854:
12848:
12847:
12846:
12836:
12834:
12831:
12830:
12798:
12792:
12791:
12790:
12783:
12771:
12766:
12765:
12759:
12754:
12753:
12752:
12746:
12745:
12744:
12737:
12735:
12723:
12717:
12716:
12715:
12707:
12704:
12703:
12671:
12665:
12664:
12663:
12654:
12649:
12648:
12640:
12635:
12634:
12633:
12627:
12626:
12625:
12616:
12610:
12609:
12608:
12600:
12597:
12596:
12580:
12577:
12576:
12559:
12554:
12553:
12545:
12542:
12541:
12538:
12517:
12514:
12513:
12492:
12488:
12486:
12483:
12482:
12456:
12452:
12434:
12430:
12418:
12412:
12411:
12410:
12401:
12397:
12385:
12381:
12379:
12376:
12375:
12351:
12345:
12344:
12343:
12334:
12330:
12328:
12325:
12324:
12308:
12307:
12301:
12297:
12285:
12281:
12263:
12262:
12251:
12247:
12235:
12231:
12212:
12208:
12197:
12193:
12181:
12177:
12158:
12154:
12145:
12144:
12128:
12121:
12117:
12105:
12101:
12082:
12080:
12073:
12064:
12058:
12057:
12056:
12046:
12044:
12041:
12040:
12023:
12018:
12017:
12015:
12012:
12011:
11992:
11988:
11976:
11972:
11945:
11941:
11908:
11902:
11901:
11900:
11885:
11879:
11878:
11877:
11869:
11866:
11865:
11839:
11835:
11833:
11830:
11829:
11813:
11810:
11809:
11775:
11769:
11768:
11767:
11758:
11753:
11752:
11744:
11739:
11738:
11737:
11731:
11730:
11729:
11720:
11714:
11713:
11712:
11704:
11701:
11700:
11674:
11668:
11667:
11666:
11664:
11661:
11660:
11643:
11638:
11637:
11635:
11632:
11631:
11624:
11595:
11590:
11589:
11584:
11580:
11574:
11573:
11572:
11563:
11558:
11557:
11549:
11544:
11543:
11542:
11536:
11535:
11534:
11525:
11519:
11518:
11517:
11509:
11506:
11505:
11488:
11483:
11482:
11474:
11471:
11470:
11467:
11465:Normal sequence
11436:
11432:
11414:
11410:
11402:
11399:
11398:
11367:
11364:
11363:
11336:
11328:
11325:
11324:
11292:
11288:
11286:
11283:
11282:
11259:
11242:
11219:
11216:
11215:
11190:
11173:
11165:
11162:
11161:
11118:
11112:
11111:
11110:
11101:
11097:
11095:
11092:
11091:
11088:Cartier divisor
11071:
11068:
11067:
11041:
11035:
11034:
11033:
11031:
11028:
11027:
10993:
10989:
10987:
10984:
10983:
10954:
10950:
10932:
10928:
10926:
10923:
10922:
10900:
10897:
10896:
10889:
10884:
10869:is the rank of
10840:
10836:
10834:
10831:
10830:
10783:
10779:
10777:
10774:
10773:
10758:
10722:
10718:
10688:
10684:
10663:
10659:
10641:
10637:
10628:
10624:
10619:
10616:
10615:
10598:
10594:
10589:
10586:
10585:
10559:
10555:
10534:
10530:
10528:
10525:
10524:
10477:
10468:
10464:
10462:
10459:
10458:
10426:
10417:
10413:
10398:
10394:
10379:
10375:
10361:
10358:
10357:
10330:
10326:
10318:
10316:
10313:
10312:
10277:
10273:
10271:
10268:
10267:
10251:
10248:
10247:
10222:
10220:
10217:
10216:
10213:
10187:
10184:
10183:
10142:
10140:
10137:
10136:
10100:
10099:
10095:
10087:
10084:
10083:
10062:
10061:
10057:
10028:
10025:
10024:
9991:
9971:
9948:
9945:
9944:
9915:
9898:
9890:
9887:
9886:
9885:Additivity: If
9870:Alternatively,
9868:
9845:
9837:
9836:
9821:
9813:
9812:
9810:
9807:
9806:
9774:
9766:
9765:
9763:
9760:
9759:
9718:
9714:
9696:
9692:
9686:
9675:
9647:
9643:
9641:
9638:
9637:
9573:
9570:
9569:
9547:
9544:
9543:
9518:
9515:
9514:
9482:
9478:
9472:
9468:
9453:
9449:
9440:
9436:
9434:
9431:
9430:
9386:
9383:
9382:
9381:Naturality: If
9343:
9339:
9337:
9334:
9333:
9327:
9285:
9281:
9263:
9259:
9241:
9237:
9220:
9217:
9216:
9183:
9165:
9161:
9159:
9156:
9155:
9148:
9137:-th Chern class
9104:
9064:
9063:
9054:
9050:
9048:
9045:
9044:
9025:
9016:
9008:
9007:
8998:
8994:
8967:
8966:
8957:
8953:
8951:
8948:
8947:
8921:
8920:
8911:
8907:
8892:
8891:
8889:
8886:
8885:
8868:
8860:
8859:
8857:
8854:
8853:
8850:
8814:
8810:
8798:
8794:
8788:
8784:
8772:
8767:
8751:
8747:
8735:
8730:
8714:
8710:
8683:
8679:
8671:
8668:
8667:
8645:
8642:
8641:
8619:
8615:
8609:
8605:
8593:
8589:
8577:
8572:
8556:
8552:
8525:
8521:
8513:
8510:
8509:
8487:
8484:
8483:
8454:
8450:
8441:
8437:
8435:
8432:
8431:
8400:
8396:
8381:
8377:
8368:
8364:
8362:
8359:
8358:
8357:In particular,
8344:
8343:
8323:
8319:
8318:
8314:
8297:
8293:
8292:
8288:
8268:
8264:
8249:
8245:
8244:
8233:
8221:
8217:
8208:
8204:
8201:
8200:
8180:
8176:
8175:
8171:
8154:
8150:
8149:
8145:
8125:
8121:
8106:
8102:
8101:
8090:
8075:
8071:
8062:
8058:
8054:
8052:
8049:
8048:
8032:
8029:
8028:
8011:
8007:
7992:
7988:
7986:
7983:
7982:
7960:
7956:
7944:
7940:
7919:
7915:
7909:
7898:
7879:
7875:
7863:
7859:
7838:
7834:
7827:
7806:
7800:
7799:
7798:
7792:
7781:
7753:
7749:
7728:
7724:
7722:
7719:
7718:
7656:
7652:
7650:
7647:
7646:
7629:
7625:
7613:
7609:
7597:
7593:
7578:
7574:
7568:
7557:
7529:
7525:
7523:
7520:
7519:
7486:
7482:
7476:
7472:
7451:
7447:
7438:
7434:
7432:
7429:
7428:
7412:
7409:
7408:
7391:
7387:
7385:
7382:
7381:
7353:
7349:
7334:
7330:
7324:
7313:
7291:
7287:
7285:
7282:
7281:
7270:
7243:
7239:
7224:
7220:
7212:
7201:
7192:
7188:
7179:
7175:
7161:
7146:
7142:
7130:
7126:
7106:
7103:
7102:
7100:
7067:
7052:
7048:
7046:
7043:
7042:
7022:
7018:
6999:
6998:
6993:
6987:
6984:
6983:
6949:
6948:
6943:
6937:
6934:
6933:
6905:
6901:
6892:
6887:
6865:
6861:
6859:
6856:
6855:
6854:One then puts:
6833:
6818:
6814:
6802:
6798:
6783:
6779:
6771:
6762:
6757:
6751:
6748:
6747:
6745:
6736:
6730:
6720:Borel's theorem
6702:
6698:
6683:
6679:
6677:
6674:
6673:
6647:
6606:
6602:
6593:
6588:
6571:
6562:
6558:
6548:
6528:
6524:
6520:
6516:
6509:
6503:
6499:
6497:
6491:
6486:
6462:
6459:
6458:
6416:
6412:
6400:
6396:
6390:
6386:
6374:
6369:
6352:
6340:
6336:
6324:
6319:
6302:
6293:
6289:
6263:
6260:
6259:
6258:from writing.)
6229:
6211:
6207:
6183:
6179:
6170:
6166:
6143:
6139:
6138:
6134:
6108:
6104:
6103:
6099:
6079:
6076:
6075:
6070:and so on (cf.
6054:
6050:
6038:
6033:
6020:
6016:
6007:
6003:
5994:
5990:
5988:
5985:
5984:
5961:
5957:
5942:
5938:
5929:
5925:
5907:
5903:
5888:
5884:
5875:
5871:
5862:
5858:
5849:
5844:
5825:
5820:
5814:
5811:
5810:
5809:
5796:
5790:
5781:
5775:
5726:
5722:
5720:
5717:
5716:
5710:
5701:
5688:
5682:
5673:
5660:
5627:
5623:
5599:
5595:
5586:
5582:
5564:
5560:
5558:
5555:
5554:
5520:
5516:
5514:
5511:
5510:
5490:
5486:
5477:
5473:
5455:
5451:
5449:
5446:
5445:
5413:
5409:
5376:
5372:
5345:
5341:
5339:
5336:
5335:
5318:
5314:
5299:
5295:
5287:
5284:
5283:
5258:
5249:
5245:
5230:
5226:
5212:
5197:
5193:
5191:
5188:
5187:
5185:
5149:
5145:
5121:
5117:
5094:
5091:
5090:
5056:
5052:
5050:
5047:
5046:
5040:
5007:
5003:
4988:
4984:
4957:
4953:
4929:
4925:
4923:
4920:
4919:
4913:
4896:
4872:
4868:
4861:
4846:
4840:
4839:
4838:
4826:
4821:
4820:
4815:
4806:
4802:
4800:
4797:
4796:
4774:
4771:
4770:
4762:is the dual of
4746:
4742:
4740:
4737:
4736:
4710:
4706:
4691:
4687:
4678:
4674:
4672:
4669:
4668:
4637:
4632:
4631:
4626:
4625:
4619:
4618:
4617:
4615:
4612:
4611:
4592:
4583:
4578:
4577:
4572:
4563:
4559:
4557:
4554:
4553:
4523:
4519:
4492:
4488:
4471:
4466:
4465:
4460:
4459:
4453:
4452:
4451:
4433:
4425:
4424:
4402:
4397:
4388:
4383:
4382:
4377:
4369:
4366:
4365:
4342:
4338:
4329:
4325:
4311:
4308:
4307:
4304:
4268:
4264:
4252:
4251:
4227:
4226:
4205:
4204:
4165:
4164:
4146:
4145:
4136:
4131:
4130:
4125:
4120:
4117:
4116:
4084:
4078:
4077:
4076:
4049:
4048:
4046:
4043:
4042:
4004:
4003:
3992:
3989:
3988:
3968:
3963:
3962:
3957:
3952:
3949:
3948:
3923:
3918:
3917:
3912:
3910:
3907:
3906:
3879:
3874:
3873:
3871:
3868:
3867:
3843:
3839:
3837:
3834:
3833:
3793:
3787:
3786:
3785:
3776:
3772:
3763:
3759:
3758:
3753:
3747:
3742:
3741:
3723:
3722:
3710:
3705:
3692:
3688:
3679:
3675:
3671:
3666:
3660:
3655:
3654:
3646:
3641:
3640:
3635:
3634:
3630:
3622:
3619:
3618:
3598:
3594:
3589:
3583:
3579:
3571:
3568:
3567:
3550:
3544:
3543:
3542:
3540:
3537:
3536:
3519:
3514:
3513:
3505:
3500:
3499:
3494:
3493:
3489:
3487:
3484:
3483:
3481:cotangent sheaf
3448:
3443:
3432:
3428:
3419:
3415:
3406:
3402:
3393:
3389:
3388:
3386:
3374:
3370:
3365:
3359:
3355:
3343:
3339:
3337:
3334:
3333:
3306:
3302:
3290:
3282:
3281:
3273:
3270:
3269:
3252:
3247:
3246:
3241:
3214:
3209:
3208:
3200:
3197:
3196:
3170:
3165:
3164:
3162:
3159:
3158:
3141:
3137:
3122:
3118:
3116:
3113:
3112:
3064:
3056:
3055:
3054:
3048:
3047:
3046:
3044:
3041:
3040:
3021:
3013:
3012:
3011:
3005:
3004:
3003:
3001:
2998:
2997:
2974:
2966:
2965:
2938:
2934:
2920:
2912:
2911:
2910:
2904:
2903:
2902:
2891:
2883:
2882:
2881:
2875:
2874:
2873:
2865:
2862:
2861:
2858:
2836:
2828:
2827:
2822:
2819:
2818:
2808:Stokes' theorem
2777:
2773:
2767:
2762:
2761:
2753:
2743:
2731:
2730:
2717:
2715:
2702:
2693:
2689:
2684:
2681:
2680:
2640:
2635:
2634:
2630:
2621:
2617:
2615:
2612:
2611:
2586:
2582:
2576:
2571:
2570:
2562:
2552:
2540:
2539:
2523:
2521:
2513:
2510:
2509:
2484:
2480:
2474:
2469:
2468:
2460:
2450:
2438:
2437:
2427:
2425:
2417:
2414:
2413:
2376:
2372:
2370:
2367:
2366:
2339:
2330:
2322:
2321:
2312:
2308:
2306:
2303:
2302:
2259:
2251:
2248:
2247:
2230:
2222:
2221:
2210:
2207:
2206:
2194:. Suppose that
2170:
2162:
2161:
2159:
2156:
2155:
2152:
2147:
2116:
2115:
2104:
2098:
2097:
2086:
2076:
2075:
2071:
2062:
2061:
2050:
2039:
2030:
2026:
2017:
2010:
2001:
2000:
1995:
1990:
1989:
1982:
1981:
1963:
1959:
1957:
1954:
1953:
1922:
1919:
1918:
1901:
1892:
1891:
1886:
1880:
1877:
1876:
1848:
1837:
1825:
1821:
1812:
1803:
1802:
1797:
1792:
1787:
1779:
1761:
1757:
1749:
1746:
1745:
1720:
1707:
1706:
1701:
1700:
1695:
1692:
1691:
1664:
1662:
1659:
1658:
1607:
1596:
1594:
1591:
1590:
1559:
1557:
1554:
1553:
1510:
1507:
1506:
1483:
1471:
1467:
1465:
1462:
1461:
1447:
1412:
1408:
1399:
1395:
1391:
1384:
1380:
1366:
1346:
1337:
1333:
1322:
1307:
1303:
1292:
1285:
1283:
1271:
1267:
1258:
1254:
1250:
1243:
1239:
1225:
1213:
1209:
1198:
1197:
1195:
1179:
1161:
1160:
1158:
1148:
1144:
1135:
1131:
1116:
1112:
1106:
1100:
1097:
1096:
1060:
1057:
1056:
1009:
983:
980:
979:
912:
910:
907:
906:
886:addition of an
873:identity matrix
826:connection form
789:
772:
769:
768:
748:
745:
744:
711:
708:
707:
688:
684:
669:
665:
659:
631:
620:
618:
617:
613:
608:
605:
604:
582:
579:
578:
541:
537:
535:
532:
531:
521:smooth manifold
500:
494:
489:
437:
428:
424:
406:
402:
388:
373:
369:
367:
364:
363:
344:
329:
325:
323:
320:
319:
303:of the bundle.
270:
263:
171:Schubert cycles
131:homotopy theory
124:
85:
80:
17:
12:
11:
5:
16245:
16235:
16234:
16229:
16212:
16211:
16209:
16208:
16198:
16197:
16196:
16191:
16186:
16171:
16161:
16151:
16139:
16128:
16125:
16124:
16122:
16121:
16116:
16111:
16106:
16101:
16096:
16090:
16088:
16082:
16081:
16079:
16078:
16073:
16068:
16066:Winding number
16063:
16058:
16052:
16050:
16046:
16045:
16043:
16042:
16037:
16032:
16027:
16022:
16017:
16012:
16007:
16006:
16005:
16000:
15998:homotopy group
15990:
15989:
15988:
15983:
15978:
15973:
15968:
15958:
15953:
15948:
15938:
15936:
15932:
15931:
15924:
15922:
15920:
15919:
15914:
15909:
15908:
15907:
15897:
15896:
15895:
15885:
15880:
15875:
15870:
15865:
15859:
15857:
15853:
15852:
15845:
15844:
15837:
15830:
15822:
15816:
15815:
15806:
15794:
15793:External links
15791:
15790:
15789:
15784:
15771:
15766:
15746:
15741:
15723:
15717:
15697:
15691:
15676:
15631:
15625:
15610:
15567:
15564:
15561:
15560:
15546:Consequently,
15531:
15526:
15522:
15518:
15515:
15512:
15507:
15503:
15499:
15494:
15490:
15484:
15479:
15476:
15473:
15469:
15465:
15462:
15460:
15458:
15455:
15450:
15446:
15442:
15439:
15436:
15431:
15426:
15423:
15420:
15416:
15412:
15409:
15407:
15405:
15402:
15397:
15393:
15389:
15386:
15381:
15376:
15373:
15370:
15366:
15362:
15359:
15357:
15355:
15352:
15347:
15343:
15339:
15336:
15333:
15328:
15324:
15320:
15317:
15314:
15311:
15309:
15307:
15304:
15301:
15298:
15295:
15294:
15274:
15271:
15268:
15265:
15260:
15256:
15250:
15245:
15242:
15239:
15235:
15231:
15228:
15225:
15222:
15219:
15199:
15196:
15191:
15187:
15183:
15180:
15177:
15172:
15168:
15164:
15159:
15155:
15151:
15148:
15145:
15142:
15137:
15133:
15110:
15106:
15073:
15058:
15054:
15031:
15027:
15010:
14998:
14986:
14974:
14962:
14939:
14935:
14930:
14926:
14922:
14917:
14913:
14910:
14907:
14904:
14901:
14898:
14895:
14892:
14889:
14885:
14881:
14878:
14875:
14870:
14867:
14863:
14859:
14856:
14851:
14846:
14842:
14838:
14835:
14831:
14827:
14824:
14821:
14816:
14813:
14809:
14792:
14780:
14776:Euler sequence
14767:
14758:
14746:Hatcher, Allen
14737:
14730:
14708:
14707:
14705:
14702:
14701:
14700:
14695:
14690:
14685:
14680:
14675:
14670:
14663:
14660:
14651:
14648:
14573:tangent bundle
14553:
14550:
14549:
14548:
14523:For varieties
14521:
14510:
14502:
14499:
14480:
14477:
14461:
14437:
14434:
14412:
14408:
14385:
14381:
14375:
14371:
14348:
14343:
14339:
14310:
14290:
14287:
14267:
14264:
14248:
14245:
14207:abelian groups
14184:
14181:
14178:
14175:
14172:
14169:
14166:
14163:
14160:
14157:
14154:
14151:
14148:
14145:
14142:
14139:
14136:
14133:
14130:
14111:
14108:
14105:
14102:
14099:
14096:
14093:
14090:
14087:
14084:
14081:
14078:
14075:
14072:
14069:
14066:
14063:
14060:
14057:
14023:
14018:
14013:
14007:
14004:
13999:
13996:
13990:
13986:
13983:
13979:
13975:
13972:
13968:
13964:
13961:
13958:
13955:
13952:
13949:
13908:
13905:
13902:
13899:
13896:
13893:
13890:
13885:
13881:
13877:
13874:
13871:
13868:
13865:
13860:
13856:
13852:
13849:
13846:
13841:
13837:
13833:
13830:
13825:
13821:
13817:
13814:
13809:
13805:
13801:
13796:
13793:
13788:
13785:
13782:
13779:
13776:
13771:
13767:
13763:
13760:
13755:
13751:
13747:
13744:
13739:
13735:
13731:
13726:
13723:
13718:
13715:
13712:
13709:
13704:
13700:
13696:
13693:
13690:
13687:
13684:
13681:
13678:
13675:
13672:
13669:
13666:
13663:
13639:
13636:
13631:
13626:
13622:
13618:
13615:
13612:
13607:
13602:
13598:
13594:
13588:
13585:
13581:
13574:
13569:
13566:
13563:
13559:
13555:
13548:
13544:
13539:
13535:
13532:
13529:
13522:
13518:
13513:
13509:
13506:
13503:
13500:
13497:
13494:
13474:
13471:
13466:
13462:
13458:
13453:
13449:
13445:
13440:
13436:
13413:
13409:
13405:
13402:
13399:
13394:
13390:
13386:
13383:
13361:
13355:
13352:
13345:
13341:
13337:
13334:
13329:
13325:
13316:
13311:
13308:
13305:
13301:
13297:
13294:
13291:
13288:
13285:
13280:
13276:
13272:
13269:
13266:
13263:
13260:
13257:
13254:
13251:
13248:
13226:
13223:
13221:
13218:
13201:
13198:
13178:
13175:
13172:
13152:
13126:
13122:
13118:
13115:
13112:
13107:
13103:
13099:
13096:
13093:
13090:
13087:
13084:
13081:
13078:
13075:
13072:
13069:
13066:
13063:
13060:
13057:
13054:
13051:
13048:
13045:
13043:
13041:
13038:
13033:
13029:
13025:
13022:
13017:
13013:
13009:
13006:
13003:
13000:
12997:
12994:
12991:
12988:
12985:
12980:
12976:
12972:
12969:
12966:
12963:
12960:
12957:
12954:
12951:
12948:
12945:
12942:
12939:
12936:
12934:
12932:
12926:
12923:
12920:
12917:
12914:
12911:
12908:
12905:
12898:
12894:
12890:
12887:
12884:
12881:
12878:
12875:
12869:
12866:
12864:
12862:
12857:
12851:
12845:
12842:
12839:
12838:
12815:
12812:
12809:
12806:
12801:
12795:
12789:
12786:
12781:
12774:
12769:
12762:
12757:
12749:
12743:
12740:
12734:
12731:
12726:
12720:
12714:
12711:
12691:
12688:
12685:
12682:
12679:
12674:
12668:
12662:
12657:
12652:
12643:
12638:
12630:
12624:
12619:
12613:
12607:
12604:
12584:
12562:
12557:
12552:
12549:
12537:
12534:
12521:
12495:
12491:
12470:
12467:
12464:
12459:
12455:
12451:
12448:
12443:
12440:
12437:
12433:
12429:
12426:
12421:
12415:
12409:
12404:
12400:
12394:
12391:
12388:
12384:
12359:
12354:
12348:
12342:
12337:
12333:
12304:
12300:
12296:
12293:
12288:
12284:
12280:
12277:
12274:
12271:
12268:
12266:
12264:
12260:
12254:
12250:
12246:
12243:
12238:
12234:
12230:
12227:
12224:
12221:
12218:
12215:
12211:
12206:
12200:
12196:
12192:
12189:
12184:
12180:
12176:
12173:
12170:
12167:
12164:
12161:
12157:
12153:
12150:
12148:
12146:
12140:
12137:
12134:
12131:
12124:
12120:
12116:
12113:
12108:
12104:
12100:
12097:
12094:
12091:
12088:
12085:
12079:
12076:
12074:
12072:
12067:
12061:
12055:
12052:
12049:
12048:
12026:
12021:
11995:
11991:
11987:
11984:
11979:
11975:
11971:
11968:
11965:
11962:
11959:
11956:
11953:
11948:
11944:
11940:
11937:
11934:
11931:
11928:
11925:
11922:
11919:
11916:
11911:
11905:
11899:
11896:
11893:
11888:
11882:
11876:
11873:
11853:
11850:
11847:
11842:
11838:
11817:
11795:
11792:
11789:
11786:
11783:
11778:
11772:
11766:
11761:
11756:
11747:
11742:
11734:
11728:
11723:
11717:
11711:
11708:
11688:
11685:
11682:
11677:
11671:
11646:
11641:
11623:
11620:
11608:
11605:
11598:
11593:
11587:
11583:
11577:
11571:
11566:
11561:
11552:
11547:
11539:
11533:
11528:
11522:
11516:
11513:
11491:
11486:
11481:
11478:
11466:
11463:
11462:
11461:
11450:
11447:
11444:
11439:
11435:
11431:
11428:
11425:
11422:
11417:
11413:
11409:
11406:
11386:
11383:
11380:
11377:
11374:
11371:
11360:
11349:
11346:
11343:
11335:
11332:
11312:
11309:
11306:
11303:
11300:
11295:
11291:
11280:
11269:
11265:
11262:
11258:
11255:
11252:
11248:
11245:
11241:
11238:
11235:
11232:
11229:
11226:
11223:
11203:
11200:
11196:
11193:
11189:
11186:
11183:
11179:
11176:
11172:
11169:
11158:
11147:
11144:
11141:
11138:
11135:
11132:
11129:
11126:
11121:
11115:
11109:
11104:
11100:
11075:
11055:
11052:
11049:
11044:
11038:
11024:
11013:
11010:
11007:
11004:
11001:
10996:
10992:
10968:
10965:
10962:
10957:
10953:
10949:
10946:
10943:
10940:
10935:
10931:
10910:
10907:
10904:
10888:
10885:
10883:
10880:
10879:
10878:
10854:
10851:
10848:
10843:
10839:
10823:
10803:
10800:
10797:
10794:
10791:
10786:
10782:
10757:
10754:
10739:
10736:
10733:
10730:
10725:
10721:
10717:
10714:
10711:
10708:
10705:
10702:
10697:
10694:
10691:
10687:
10683:
10680:
10677:
10672:
10669:
10666:
10662:
10658:
10655:
10652:
10649:
10644:
10640:
10636:
10631:
10627:
10623:
10601:
10597:
10593:
10573:
10570:
10567:
10562:
10558:
10554:
10551:
10548:
10545:
10542:
10537:
10533:
10493:
10490:
10487:
10484:
10480:
10476:
10471:
10467:
10442:
10439:
10436:
10433:
10429:
10425:
10420:
10416:
10412:
10407:
10404:
10401:
10397:
10393:
10390:
10387:
10382:
10378:
10374:
10371:
10368:
10365:
10338:
10333:
10329:
10325:
10321:
10300:
10297:
10294:
10291:
10288:
10285:
10280:
10276:
10255:
10235:
10232:
10229:
10225:
10211:
10197:
10194:
10191:
10155:
10152:
10149:
10145:
10126:
10125:
10109:
10103:
10098:
10094:
10091:
10071:
10065:
10060:
10056:
10053:
10050:
10047:
10044:
10041:
10038:
10035:
10032:
10013:
10001:
9997:
9994:
9990:
9987:
9984:
9981:
9977:
9974:
9970:
9967:
9964:
9961:
9958:
9955:
9952:
9941:exact sequence
9928:
9925:
9921:
9918:
9914:
9911:
9908:
9904:
9901:
9897:
9894:
9883:
9867:
9864:
9863:
9862:
9848:
9843:
9840:
9835:
9830:
9827:
9824:
9819:
9816:
9777:
9772:
9769:
9752:
9741:
9738:
9735:
9732:
9727:
9724:
9721:
9717:
9713:
9710:
9707:
9704:
9699:
9695:
9689:
9684:
9681:
9678:
9674:
9670:
9667:
9664:
9661:
9658:
9655:
9650:
9646:
9625:
9622:
9619:
9616:
9613:
9610:
9607:
9604:
9601:
9598:
9595:
9592:
9589:
9586:
9583:
9580:
9577:
9557:
9554:
9551:
9528:
9525:
9522:
9508:
9496:
9493:
9490:
9485:
9481:
9475:
9471:
9467:
9464:
9461:
9456:
9452:
9448:
9443:
9439:
9402:
9399:
9396:
9393:
9390:
9379:
9363:
9360:
9357:
9354:
9351:
9346:
9342:
9326:
9323:
9308:
9305:
9302:
9299:
9296:
9293:
9288:
9284:
9280:
9277:
9274:
9271:
9266:
9262:
9258:
9255:
9252:
9249:
9244:
9240:
9236:
9233:
9230:
9227:
9224:
9193:
9190:
9186:
9182:
9179:
9176:
9171:
9168:
9164:
9146:
9103:
9100:
9087:
9084:
9081:
9078:
9075:
9072:
9067:
9062:
9057:
9053:
9032:
9028:
9024:
9019:
9014:
9011:
9006:
9001:
8997:
8993:
8990:
8987:
8984:
8981:
8978:
8975:
8970:
8965:
8960:
8956:
8935:
8932:
8929:
8924:
8919:
8914:
8910:
8906:
8903:
8900:
8895:
8884:. Recall that
8871:
8866:
8863:
8849:
8846:
8845:
8844:
8836:
8835:
8834:
8833:
8822:
8817:
8813:
8809:
8806:
8801:
8797:
8791:
8787:
8783:
8780:
8775:
8770:
8766:
8762:
8759:
8754:
8750:
8746:
8743:
8738:
8733:
8729:
8725:
8722:
8717:
8713:
8709:
8706:
8703:
8700:
8697:
8694:
8691:
8686:
8682:
8678:
8675:
8655:
8652:
8649:
8638:
8627:
8622:
8618:
8612:
8608:
8604:
8601:
8596:
8592:
8588:
8585:
8580:
8575:
8571:
8567:
8564:
8559:
8555:
8551:
8548:
8545:
8542:
8539:
8536:
8533:
8528:
8524:
8520:
8517:
8497:
8494:
8491:
8468:
8465:
8462:
8457:
8453:
8449:
8444:
8440:
8428:
8417:
8414:
8411:
8408:
8403:
8399:
8395:
8392:
8389:
8384:
8380:
8376:
8371:
8367:
8342:
8339:
8336:
8333:
8326:
8322:
8317:
8313:
8310:
8307:
8300:
8296:
8291:
8287:
8284:
8281:
8278:
8271:
8267:
8263:
8260:
8257:
8252:
8248:
8243:
8239:
8236:
8234:
8232:
8229:
8224:
8220:
8216:
8211:
8207:
8203:
8202:
8199:
8196:
8193:
8190:
8183:
8179:
8174:
8170:
8167:
8164:
8157:
8153:
8148:
8144:
8141:
8138:
8135:
8128:
8124:
8120:
8117:
8114:
8109:
8105:
8100:
8096:
8093:
8091:
8089:
8086:
8083:
8078:
8074:
8070:
8065:
8061:
8057:
8056:
8036:
8014:
8010:
8006:
8003:
8000:
7995:
7991:
7979:
7968:
7963:
7959:
7955:
7952:
7947:
7943:
7939:
7936:
7933:
7928:
7925:
7922:
7918:
7912:
7907:
7904:
7901:
7897:
7893:
7890:
7887:
7882:
7878:
7874:
7871:
7866:
7862:
7858:
7855:
7852:
7847:
7844:
7841:
7837:
7830:
7825:
7821:
7818:
7815:
7812:
7809:
7803:
7795:
7790:
7787:
7784:
7780:
7776:
7773:
7770:
7767:
7764:
7761:
7756:
7752:
7748:
7745:
7742:
7739:
7736:
7731:
7727:
7706:
7703:
7700:
7697:
7694:
7691:
7688:
7685:
7682:
7679:
7676:
7673:
7670:
7667:
7664:
7659:
7655:
7632:
7628:
7622:
7619:
7616:
7612:
7608:
7605:
7600:
7596:
7592:
7589:
7586:
7581:
7577:
7571:
7566:
7563:
7560:
7556:
7552:
7549:
7546:
7543:
7540:
7537:
7532:
7528:
7512:
7500:
7497:
7494:
7489:
7485:
7479:
7475:
7471:
7468:
7465:
7462:
7459:
7454:
7450:
7446:
7441:
7437:
7416:
7394:
7390:
7356:
7352:
7348:
7345:
7342:
7337:
7333:
7327:
7322:
7319:
7316:
7312:
7308:
7305:
7302:
7299:
7294:
7290:
7269:
7266:
7254:
7251:
7246:
7242:
7238:
7235:
7232:
7227:
7223:
7219:
7215:
7211:
7208:
7204:
7200:
7195:
7191:
7187:
7182:
7178:
7174:
7171:
7168:
7164:
7160:
7157:
7154:
7149:
7145:
7141:
7138:
7133:
7129:
7125:
7122:
7119:
7116:
7113:
7110:
7096:
7090:Yoneda's lemma
7077:
7074:
7070:
7066:
7063:
7060:
7055:
7051:
7030:
7025:
7021:
7017:
7014:
7011:
7008:
7002:
6996:
6992:
6952:
6946:
6942:
6916:
6913:
6908:
6904:
6900:
6895:
6890:
6886:
6882:
6879:
6876:
6873:
6868:
6864:
6843:
6840:
6836:
6832:
6829:
6826:
6821:
6817:
6813:
6810:
6805:
6801:
6797:
6794:
6791:
6786:
6782:
6778:
6774:
6770:
6765:
6760:
6756:
6741:
6732:
6726:
6705:
6701:
6697:
6694:
6691:
6686:
6682:
6643:
6623:
6620:
6617:
6614:
6609:
6605:
6601:
6596:
6591:
6587:
6583:
6578:
6575:
6570:
6565:
6561:
6555:
6552:
6547:
6544:
6541:
6531:
6527:
6523:
6519:
6515:
6512:
6506:
6502:
6494:
6489:
6485:
6481:
6478:
6475:
6472:
6469:
6466:
6433:
6430:
6427:
6424:
6419:
6415:
6411:
6408:
6403:
6399:
6393:
6389:
6385:
6382:
6377:
6372:
6368:
6364:
6359:
6356:
6351:
6348:
6343:
6339:
6335:
6332:
6327:
6322:
6318:
6314:
6309:
6306:
6301:
6296:
6292:
6288:
6285:
6282:
6279:
6276:
6273:
6270:
6267:
6239:
6236:
6232:
6228:
6225:
6222:
6219:
6214:
6210:
6206:
6203:
6200:
6197:
6194:
6191:
6186:
6182:
6178:
6173:
6169:
6165:
6162:
6157:
6154:
6151:
6146:
6142:
6137:
6133:
6130:
6127:
6122:
6119:
6116:
6111:
6107:
6102:
6098:
6095:
6092:
6089:
6086:
6083:
6057:
6053:
6049:
6046:
6041:
6036:
6032:
6028:
6023:
6019:
6015:
6010:
6006:
6002:
5997:
5993:
5972:
5969:
5964:
5960:
5956:
5953:
5950:
5945:
5941:
5937:
5932:
5928:
5924:
5921:
5918:
5915:
5910:
5906:
5902:
5899:
5896:
5891:
5887:
5883:
5878:
5874:
5870:
5865:
5861:
5857:
5852:
5847:
5843:
5839:
5836:
5833:
5828:
5823:
5819:
5805:
5786:
5777:
5771:
5757:
5740:
5737:
5734:
5729:
5725:
5711:'s. Either by
5706:
5697:
5684:
5678:
5669:
5656:
5644:
5641:
5638:
5635:
5630:
5626:
5622:
5619:
5616:
5613:
5610:
5607:
5602:
5598:
5594:
5589:
5585:
5581:
5578:
5575:
5572:
5567:
5563:
5534:
5531:
5528:
5523:
5519:
5498:
5493:
5489:
5485:
5480:
5476:
5472:
5469:
5466:
5463:
5458:
5454:
5433:
5430:
5427:
5424:
5421:
5416:
5412:
5408:
5405:
5402:
5399:
5396:
5393:
5390:
5387:
5384:
5379:
5375:
5371:
5368:
5365:
5362:
5359:
5356:
5353:
5348:
5344:
5321:
5317:
5313:
5310:
5307:
5302:
5298:
5294:
5291:
5271:
5268:
5264:
5261:
5257:
5252:
5248:
5244:
5241:
5238:
5233:
5229:
5225:
5222:
5218:
5215:
5211:
5208:
5205:
5200:
5196:
5181:
5163:
5160:
5157:
5152:
5148:
5144:
5141:
5138:
5135:
5132:
5129:
5124:
5120:
5116:
5113:
5110:
5107:
5104:
5101:
5098:
5070:
5067:
5064:
5059:
5055:
5036:
5015:
5010:
5006:
5002:
4999:
4996:
4991:
4987:
4983:
4980:
4977:
4974:
4971:
4968:
4965:
4960:
4956:
4952:
4949:
4946:
4943:
4940:
4937:
4932:
4928:
4909:
4895:
4892:
4880:
4875:
4871:
4864:
4859:
4855:
4852:
4849:
4843:
4837:
4834:
4829:
4824:
4818:
4814:
4809:
4805:
4784:
4781:
4778:
4749:
4745:
4724:
4721:
4718:
4713:
4709:
4705:
4702:
4699:
4694:
4690:
4686:
4681:
4677:
4656:
4653:
4650:
4647:
4640:
4635:
4629:
4622:
4599:
4595:
4591:
4586:
4581:
4575:
4571:
4566:
4562:
4537:
4532:
4529:
4526:
4522:
4518:
4515:
4512:
4509:
4506:
4501:
4498:
4495:
4491:
4487:
4484:
4481:
4474:
4469:
4463:
4456:
4450:
4447:
4444:
4441:
4436:
4431:
4428:
4423:
4420:
4417:
4411:
4408:
4405:
4401:
4396:
4391:
4386:
4380:
4376:
4373:
4353:
4350:
4345:
4341:
4337:
4332:
4328:
4324:
4321:
4318:
4315:
4303:
4302:
4291:
4286:
4283:
4280:
4277:
4274:
4271:
4267:
4263:
4260:
4255:
4250:
4247:
4244:
4241:
4238:
4235:
4230:
4225:
4222:
4219:
4216:
4213:
4208:
4203:
4200:
4197:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4168:
4163:
4160:
4157:
4154:
4149:
4144:
4139:
4134:
4128:
4124:
4115:. It follows:
4102:
4099:
4096:
4093:
4090:
4087:
4081:
4075:
4072:
4069:
4066:
4063:
4060:
4057:
4052:
4030:
4027:
4024:
4021:
4018:
4015:
4012:
4007:
4002:
3999:
3996:
3971:
3966:
3960:
3956:
3926:
3921:
3915:
3888:
3885:
3882:
3877:
3860:
3846:
3842:
3820:
3817:
3814:
3811:
3807:
3804:
3801:
3796:
3790:
3779:
3775:
3771:
3766:
3762:
3757:
3750:
3745:
3740:
3737:
3734:
3731:
3726:
3719:
3716:
3713:
3708:
3704:
3695:
3691:
3687:
3682:
3678:
3674:
3670:
3663:
3658:
3649:
3644:
3638:
3633:
3629:
3626:
3606:
3601:
3597:
3592:
3586:
3582:
3578:
3575:
3553:
3547:
3522:
3517:
3508:
3503:
3497:
3492:
3468:
3465:
3462:
3458:
3451:
3446:
3442:
3435:
3431:
3427:
3422:
3418:
3414:
3409:
3405:
3401:
3396:
3392:
3385:
3382:
3377:
3373:
3368:
3362:
3358:
3354:
3351:
3346:
3342:
3320:
3317:
3314:
3309:
3305:
3301:
3298:
3293:
3288:
3285:
3280:
3277:
3255:
3250:
3244:
3240:
3237:
3234:
3231:
3228:
3223:
3220:
3217:
3212:
3207:
3204:
3184:
3179:
3176:
3173:
3168:
3144:
3140:
3136:
3133:
3130:
3125:
3121:
3108:
3080:
3077:
3074:
3067:
3062:
3059:
3051:
3024:
3019:
3016:
3008:
2985:
2982:
2977:
2972:
2969:
2964:
2961:
2956:
2953:
2950:
2947:
2944:
2941:
2937:
2933:
2930:
2923:
2918:
2915:
2907:
2901:
2894:
2889:
2886:
2878:
2872:
2869:
2857:
2854:
2839:
2834:
2831:
2826:
2791:
2788:
2780:
2776:
2770:
2765:
2760:
2756:
2752:
2749:
2746:
2738:
2735:
2729:
2726:
2723:
2720:
2714:
2709:
2706:
2701:
2696:
2692:
2688:
2666:
2662:
2658:
2655:
2652:
2646:
2643:
2639:
2633:
2629:
2624:
2620:
2597:
2589:
2585:
2579:
2574:
2569:
2565:
2561:
2558:
2555:
2547:
2544:
2538:
2535:
2532:
2529:
2526:
2520:
2517:
2495:
2487:
2483:
2477:
2472:
2467:
2463:
2459:
2456:
2453:
2445:
2442:
2436:
2433:
2430:
2424:
2421:
2396:
2393:
2390:
2387:
2384:
2379:
2375:
2352:
2349:
2346:
2342:
2338:
2333:
2328:
2325:
2320:
2315:
2311:
2286:complex number
2269:
2266:
2262:
2258:
2255:
2233:
2228:
2225:
2220:
2217:
2214:
2188:Riemann sphere
2173:
2168:
2165:
2151:
2148:
2146:
2143:
2119:
2114:
2111:
2108:
2105:
2103:
2100:
2099:
2096:
2093:
2090:
2087:
2085:
2079:
2074:
2070:
2067:
2064:
2063:
2060:
2057:
2054:
2051:
2049:
2045:
2042:
2038:
2033:
2029:
2023:
2020:
2013:
2007:
2004:
1998:
1993:
1988:
1987:
1985:
1980:
1977:
1974:
1971:
1966:
1962:
1941:
1938:
1935:
1932:
1929:
1926:
1904:
1898:
1895:
1889:
1884:
1864:
1861:
1858:
1855:
1851:
1847:
1843:
1840:
1836:
1833:
1828:
1824:
1815:
1809:
1806:
1800:
1795:
1791:
1786:
1782:
1778:
1775:
1772:
1769:
1764:
1760:
1756:
1753:
1733:
1730:
1726:
1723:
1719:
1713:
1710:
1704:
1699:
1688:Gysin sequence
1670:
1667:
1613:
1610:
1606:
1602:
1599:
1578:
1575:
1572:
1569:
1565:
1562:
1544:. Thinking of
1526:
1523:
1520:
1517:
1514:
1490:
1486:
1482:
1479:
1474:
1470:
1446:
1443:
1431:
1427:
1423:
1420:
1415:
1411:
1402:
1398:
1394:
1387:
1383:
1379:
1376:
1372:
1369:
1365:
1362:
1359:
1356:
1352:
1349:
1345:
1340:
1336:
1332:
1328:
1325:
1321:
1318:
1315:
1310:
1306:
1302:
1298:
1295:
1291:
1288:
1282:
1279:
1274:
1270:
1261:
1257:
1253:
1246:
1242:
1238:
1235:
1231:
1228:
1224:
1221:
1216:
1212:
1208:
1204:
1201:
1194:
1191:
1185:
1182:
1177:
1174:
1171:
1167:
1164:
1157:
1154:
1151:
1147:
1143:
1138:
1134:
1130:
1127:
1124:
1119:
1115:
1109:
1105:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1015:
1012:
1008:
1005:
1002:
999:
996:
993:
990:
987:
967:
964:
961:
958:
955:
952:
949:
946:
943:
940:
937:
934:
931:
928:
925:
922:
918:
915:
880:representative
848:. The scalar
813:
810:
807:
804:
801:
796:
793:
788:
785:
782:
779:
776:
767:is defined as
752:
742:curvature form
721:
718:
715:
691:
687:
683:
680:
677:
672:
668:
662:
658:
654:
650:
646:
643:
637:
634:
629:
626:
623:
616:
612:
586:
576:curvature form
555:
552:
549:
544:
540:
496:Main article:
493:
490:
488:
485:
463:tensor product
450:
447:
443:
440:
436:
431:
427:
423:
420:
417:
414:
409:
405:
401:
398:
394:
391:
387:
384:
381:
376:
372:
351:
347:
343:
340:
337:
332:
328:
262:
259:
123:
120:
116:curvature form
84:
81:
79:
76:
48:vector bundles
15:
9:
6:
4:
3:
2:
16244:
16233:
16230:
16228:
16225:
16224:
16222:
16207:
16199:
16195:
16192:
16190:
16187:
16185:
16182:
16181:
16180:
16172:
16170:
16166:
16162:
16160:
16156:
16152:
16150:
16145:
16140:
16138:
16130:
16129:
16126:
16120:
16117:
16115:
16112:
16110:
16107:
16105:
16102:
16100:
16097:
16095:
16092:
16091:
16089:
16087:
16083:
16077:
16076:Orientability
16074:
16072:
16069:
16067:
16064:
16062:
16059:
16057:
16054:
16053:
16051:
16047:
16041:
16038:
16036:
16033:
16031:
16028:
16026:
16023:
16021:
16018:
16016:
16013:
16011:
16008:
16004:
16001:
15999:
15996:
15995:
15994:
15991:
15987:
15984:
15982:
15979:
15977:
15974:
15972:
15969:
15967:
15964:
15963:
15962:
15959:
15957:
15954:
15952:
15949:
15947:
15943:
15940:
15939:
15937:
15933:
15928:
15918:
15915:
15913:
15912:Set-theoretic
15910:
15906:
15903:
15902:
15901:
15898:
15894:
15891:
15890:
15889:
15886:
15884:
15881:
15879:
15876:
15874:
15873:Combinatorial
15871:
15869:
15866:
15864:
15861:
15860:
15858:
15854:
15850:
15843:
15838:
15836:
15831:
15829:
15824:
15823:
15820:
15814:
15810:
15807:
15804:
15803:Allen Hatcher
15800:
15797:
15796:
15787:
15781:
15777:
15772:
15769:
15763:
15759:
15755:
15751:
15747:
15744:
15742:9780226511832
15738:
15734:
15733:
15728:
15727:May, J. Peter
15724:
15720:
15714:
15710:
15706:
15702:
15698:
15694:
15688:
15684:
15683:
15677:
15674:
15670:
15666:
15662:
15657:
15652:
15648:
15644:
15640:
15636:
15632:
15628:
15622:
15618:
15617:
15611:
15608:
15604:
15600:
15596:
15592:
15588:
15585:(1): 85â121,
15584:
15580:
15579:
15574:
15570:
15569:
15557:
15553:
15549:
15524:
15520:
15516:
15513:
15510:
15505:
15501:
15492:
15488:
15482:
15477:
15474:
15471:
15467:
15463:
15461:
15448:
15444:
15440:
15437:
15429:
15424:
15421:
15418:
15414:
15410:
15408:
15395:
15391:
15384:
15379:
15374:
15371:
15368:
15364:
15360:
15358:
15345:
15341:
15337:
15334:
15331:
15326:
15322:
15315:
15312:
15310:
15302:
15296:
15272:
15266:
15258:
15254:
15248:
15243:
15240:
15237:
15233:
15229:
15223:
15217:
15197:
15189:
15185:
15181:
15178:
15175:
15170:
15166:
15157:
15153:
15149:
15143:
15135:
15131:
15108:
15104:
15095:
15091:
15087:
15083:
15077:
15056:
15052:
15029:
15025:
15014:
15007:
15002:
14995:
14990:
14983:
14978:
14971:
14966:
14959:
14955:
14937:
14933:
14924:
14915:
14911:
14905:
14902:
14896:
14879:
14876:
14868:
14865:
14861:
14854:
14844:
14840:
14825:
14822:
14814:
14811:
14807:
14796:
14789:
14784:
14777:
14771:
14762:
14751:
14747:
14741:
14733:
14731:3-540-90613-4
14727:
14723:
14719:
14713:
14709:
14699:
14696:
14694:
14691:
14689:
14686:
14684:
14681:
14679:
14676:
14674:
14671:
14669:
14666:
14665:
14659:
14657:
14647:
14645:
14641:
14636:
14634:
14630:
14626:
14622:
14618:
14614:
14610:
14606:
14602:
14598:
14594:
14590:
14586:
14582:
14578:
14577:Chern classes
14574:
14570:
14565:
14563:
14559:
14546:
14542:
14538:
14534:
14530:
14526:
14522:
14519:
14515:
14511:
14508:
14507:
14506:
14498:
14496:
14492:
14491:
14486:
14476:
14473:
14459:
14451:
14435:
14432:
14410:
14406:
14383:
14379:
14373:
14369:
14346:
14341:
14337:
14328:
14324:
14308:
14288:
14285:
14265:
14262:
14255:of dimension
14254:
14247:Chern numbers
14244:
14242:
14237:
14235:
14231:
14227:
14223:
14219:
14215:
14212:
14208:
14204:
14200:
14195:
14182:
14176:
14170:
14167:
14161:
14155:
14152:
14149:
14143:
14140:
14137:
14131:
14128:
14106:
14100:
14097:
14094:
14088:
14082:
14079:
14076:
14070:
14067:
14064:
14058:
14055:
14047:
14044:
14042:
14021:
14016:
14011:
14005:
14002:
13994:
13988:
13984:
13981:
13977:
13973:
13970:
13966:
13962:
13956:
13950:
13947:
13939:
13934:
13932:
13928:
13924:
13919:
13906:
13903:
13900:
13891:
13883:
13879:
13875:
13872:
13866:
13858:
13854:
13847:
13839:
13835:
13831:
13828:
13823:
13815:
13807:
13803:
13794:
13791:
13786:
13777:
13769:
13765:
13761:
13758:
13753:
13745:
13737:
13733:
13724:
13721:
13716:
13710:
13702:
13698:
13694:
13688:
13682:
13679:
13676:
13670:
13664:
13661:
13653:
13650:
13637:
13629:
13624:
13620:
13616:
13613:
13610:
13605:
13600:
13596:
13586:
13583:
13579:
13567:
13564:
13561:
13557:
13553:
13546:
13542:
13537:
13533:
13530:
13527:
13520:
13516:
13511:
13507:
13501:
13495:
13492:
13472:
13464:
13460:
13451:
13447:
13443:
13438:
13434:
13411:
13407:
13403:
13400:
13397:
13392:
13388:
13384:
13381:
13372:
13359:
13353:
13350:
13343:
13335:
13327:
13323:
13309:
13306:
13303:
13299:
13295:
13286:
13278:
13274:
13267:
13264:
13261:
13255:
13249:
13246:
13238:
13236:
13232:
13217:
13215:
13214:spin manifold
13199:
13196:
13176:
13173:
13170:
13150:
13124:
13116:
13105:
13101:
13097:
13094:
13091:
13088:
13085:
13079:
13073:
13064:
13061:
13058:
13052:
13049:
13046:
13044:
13031:
13023:
13015:
13011:
13007:
13001:
12995:
12992:
12989:
12978:
12970:
12964:
12961:
12955:
12949:
12946:
12943:
12937:
12935:
12918:
12912:
12909:
12906:
12896:
12885:
12879:
12876:
12867:
12865:
12855:
12840:
12807:
12799:
12784:
12772:
12760:
12738:
12732:
12724:
12709:
12689:
12680:
12672:
12655:
12641:
12617:
12602:
12582:
12560:
12550:
12547:
12533:
12519:
12511:
12493:
12489:
12468:
12465:
12462:
12457:
12453:
12449:
12446:
12438:
12431:
12427:
12419:
12402:
12398:
12389:
12382:
12373:
12352:
12335:
12331:
12321:
12302:
12298:
12294:
12291:
12286:
12282:
12278:
12275:
12272:
12269:
12267:
12258:
12252:
12248:
12244:
12241:
12236:
12232:
12228:
12225:
12222:
12219:
12216:
12213:
12209:
12204:
12198:
12194:
12190:
12187:
12182:
12178:
12174:
12171:
12168:
12165:
12162:
12159:
12155:
12151:
12149:
12138:
12135:
12132:
12129:
12122:
12118:
12114:
12111:
12106:
12102:
12098:
12095:
12092:
12089:
12086:
12083:
12077:
12075:
12065:
12050:
12024:
12008:
11993:
11989:
11985:
11982:
11977:
11973:
11969:
11966:
11963:
11960:
11957:
11954:
11951:
11946:
11938:
11935:
11932:
11926:
11917:
11909:
11894:
11886:
11871:
11848:
11840:
11836:
11815:
11806:
11793:
11784:
11776:
11759:
11745:
11721:
11706:
11683:
11675:
11644:
11629:
11619:
11606:
11596:
11585:
11581:
11564:
11550:
11526:
11511:
11489:
11479:
11476:
11445:
11437:
11433:
11423:
11415:
11411:
11407:
11404:
11381:
11375:
11369:
11361:
11344:
11333:
11330:
11310:
11307:
11301:
11293:
11289:
11281:
11263:
11260:
11253:
11246:
11243:
11236:
11233:
11227:
11221:
11201:
11194:
11191:
11184:
11177:
11174:
11167:
11159:
11142:
11136:
11127:
11119:
11102:
11098:
11089:
11073:
11050:
11042:
11025:
11011:
11008:
11002:
10994:
10990:
10982:
10981:
10980:
10963:
10955:
10951:
10947:
10941:
10933:
10929:
10908:
10902:
10894:
10876:
10872:
10868:
10849:
10841:
10837:
10828:
10824:
10821:
10817:
10801:
10798:
10792:
10784:
10780:
10771:
10767:
10763:
10762:
10761:
10753:
10750:
10737:
10731:
10723:
10719:
10715:
10712:
10709:
10703:
10695:
10692:
10689:
10685:
10681:
10678:
10675:
10670:
10667:
10664:
10660:
10656:
10650:
10642:
10638:
10634:
10629:
10625:
10621:
10599:
10595:
10591:
10568:
10560:
10556:
10552:
10549:
10543:
10535:
10531:
10522:
10517:
10515:
10511:
10507:
10485:
10469:
10465:
10456:
10434:
10418:
10414:
10410:
10405:
10402:
10399:
10395:
10391:
10388:
10385:
10380:
10376:
10372:
10369:
10366:
10363:
10354:
10352:
10331:
10327:
10298:
10295:
10292:
10286:
10278:
10274:
10253:
10230:
10214:
10195:
10192:
10189:
10181:
10177:
10173:
10169:
10150:
10133:
10131:
10123:
10096:
10089:
10058:
10051:
10048:
10045:
10042:
10036:
10030:
10022:
10018:
10014:
9995:
9992:
9985:
9982:
9975:
9972:
9965:
9962:
9956:
9950:
9942:
9926:
9919:
9916:
9909:
9902:
9899:
9892:
9884:
9881:
9880:
9879:
9877:
9873:
9846:
9833:
9828:
9825:
9822:
9805:
9801:
9800:Poincaré dual
9797:
9793:
9775:
9757:
9753:
9739:
9733:
9725:
9722:
9719:
9715:
9711:
9705:
9697:
9693:
9687:
9682:
9679:
9676:
9672:
9668:
9662:
9659:
9656:
9648:
9644:
9623:
9617:
9611:
9608:
9602:
9596:
9593:
9587:
9584:
9581:
9575:
9568:are given by
9555:
9552:
9549:
9542:
9526:
9520:
9512:
9509:
9491:
9483:
9479:
9473:
9469:
9465:
9459:
9454:
9450:
9441:
9437:
9428:
9424:
9420:
9416:
9400:
9394:
9391:
9388:
9380:
9377:
9361:
9358:
9352:
9344:
9340:
9332:
9331:
9330:
9322:
9319:
9306:
9303:
9300:
9294:
9286:
9282:
9278:
9272:
9264:
9260:
9256:
9250:
9242:
9238:
9234:
9228:
9222:
9215:
9211:
9207:
9191:
9180:
9177:
9169:
9166:
9162:
9153:
9149:
9142:
9138:
9136:
9131:
9127:
9123:
9119:
9116:
9112:
9109:
9099:
9085:
9082:
9073:
9055:
9051:
9022:
9017:
8999:
8995:
8991:
8988:
8985:
8976:
8958:
8954:
8930:
8917:
8912:
8904:
8901:
8869:
8842:
8838:
8837:
8820:
8815:
8811:
8807:
8804:
8799:
8795:
8789:
8785:
8781:
8778:
8773:
8768:
8764:
8760:
8757:
8752:
8748:
8744:
8741:
8736:
8731:
8727:
8723:
8720:
8715:
8711:
8707:
8704:
8701:
8698:
8692:
8689:
8684:
8680:
8673:
8653:
8650:
8647:
8639:
8625:
8620:
8616:
8610:
8606:
8602:
8599:
8594:
8590:
8586:
8583:
8578:
8573:
8569:
8565:
8562:
8557:
8553:
8549:
8546:
8543:
8540:
8534:
8531:
8526:
8522:
8515:
8495:
8492:
8489:
8481:
8480:
8463:
8455:
8451:
8447:
8442:
8438:
8429:
8415:
8409:
8401:
8397:
8393:
8387:
8382:
8378:
8369:
8365:
8340:
8334:
8324:
8320:
8315:
8311:
8308:
8305:
8298:
8294:
8289:
8282:
8279:
8269:
8265:
8261:
8258:
8255:
8250:
8246:
8241:
8237:
8235:
8227:
8222:
8218:
8209:
8205:
8197:
8191:
8181:
8177:
8172:
8168:
8165:
8162:
8155:
8151:
8146:
8139:
8136:
8126:
8122:
8118:
8115:
8112:
8107:
8103:
8098:
8094:
8092:
8084:
8081:
8076:
8072:
8063:
8059:
8034:
8012:
8008:
8004:
8001:
7998:
7993:
7989:
7980:
7966:
7961:
7953:
7945:
7941:
7934:
7926:
7923:
7920:
7916:
7910:
7905:
7902:
7899:
7895:
7891:
7888:
7885:
7880:
7872:
7864:
7860:
7853:
7845:
7842:
7839:
7835:
7823:
7819:
7816:
7813:
7810:
7807:
7793:
7788:
7785:
7782:
7778:
7774:
7771:
7768:
7762:
7754:
7750:
7746:
7743:
7737:
7729:
7725:
7704:
7701:
7698:
7695:
7692:
7689:
7686:
7683:
7680:
7677:
7671:
7668:
7665:
7657:
7653:
7630:
7626:
7620:
7617:
7614:
7606:
7598:
7594:
7587:
7579:
7575:
7569:
7564:
7561:
7558:
7554:
7550:
7544:
7541:
7538:
7530:
7526:
7517:
7513:
7495:
7487:
7483:
7477:
7469:
7466:
7460:
7452:
7448:
7439:
7435:
7414:
7392:
7388:
7380:
7376:
7375:
7374:
7372:
7354:
7350:
7343:
7335:
7331:
7325:
7320:
7317:
7314:
7310:
7306:
7300:
7292:
7288:
7279:
7275:
7265:
7252:
7244:
7240:
7236:
7233:
7230:
7225:
7221:
7209:
7198:
7193:
7189:
7180:
7176:
7172:
7158:
7155:
7147:
7143:
7139:
7131:
7127:
7123:
7120:
7111:
7108:
7099:
7095:
7091:
7075:
7064:
7061:
7053:
7049:
7023:
7019:
7015:
7012:
7006:
6994:
6990:
6981:
6977:
6973:
6969:
6944:
6940:
6931:
6927:
6914:
6906:
6902:
6893:
6888:
6884:
6880:
6874:
6866:
6862:
6841:
6830:
6827:
6819:
6815:
6803:
6799:
6795:
6792:
6789:
6784:
6780:
6768:
6763:
6758:
6754:
6744:
6740:
6735:
6729:
6725:
6721:
6703:
6699:
6692:
6689:
6684:
6680:
6671:
6667:
6663:
6659:
6655:
6651:
6646:
6642:
6638:
6634:
6621:
6618:
6615:
6607:
6603:
6599:
6594:
6589:
6585:
6576:
6573:
6568:
6563:
6559:
6553:
6550:
6545:
6542:
6539:
6529:
6525:
6521:
6517:
6513:
6510:
6504:
6500:
6492:
6487:
6483:
6479:
6473:
6467:
6464:
6457:is given by:
6456:
6452:
6448:
6444:
6431:
6428:
6425:
6417:
6413:
6409:
6406:
6401:
6397:
6391:
6387:
6383:
6380:
6375:
6370:
6366:
6357:
6354:
6349:
6341:
6337:
6333:
6330:
6325:
6320:
6316:
6307:
6304:
6299:
6294:
6290:
6286:
6283:
6280:
6274:
6268:
6265:
6257:
6253:
6237:
6234:
6230:
6220:
6212:
6208:
6204:
6201:
6198:
6192:
6184:
6180:
6171:
6167:
6163:
6160:
6152:
6144:
6140:
6135:
6131:
6128:
6125:
6117:
6109:
6105:
6100:
6096:
6090:
6084:
6081:
6073:
6055:
6051:
6047:
6044:
6039:
6034:
6030:
6026:
6021:
6017:
6013:
6008:
6004:
6000:
5995:
5991:
5962:
5958:
5954:
5951:
5948:
5943:
5939:
5930:
5926:
5922:
5919:
5916:
5908:
5904:
5900:
5897:
5894:
5889:
5885:
5876:
5872:
5863:
5859:
5855:
5850:
5845:
5841:
5837:
5834:
5831:
5826:
5821:
5817:
5808:
5804:
5800:
5794:
5789:
5785:
5780:
5774:
5769:
5765:
5761:
5756:
5754:
5735:
5727:
5723:
5714:
5709:
5705:
5700:
5696:
5692:
5687:
5681:
5677:
5672:
5668:
5664:
5659:
5636:
5628:
5624:
5620:
5617:
5614:
5608:
5600:
5596:
5587:
5583:
5579:
5573:
5565:
5561:
5552:
5548:
5545:, called the
5529:
5521:
5517:
5491:
5487:
5478:
5474:
5470:
5464:
5456:
5452:
5428:
5422:
5414:
5410:
5406:
5403:
5397:
5391:
5385:
5377:
5373:
5369:
5366:
5360:
5354:
5346:
5342:
5319:
5315:
5311:
5308:
5305:
5300:
5296:
5292:
5289:
5269:
5262:
5259:
5250:
5246:
5239:
5231:
5227:
5223:
5216:
5213:
5209:
5206:
5198:
5194:
5184:
5180:
5175:
5158:
5150:
5146:
5142:
5139:
5136:
5130:
5122:
5118:
5114:
5111:
5108:
5102:
5096:
5088:
5084:
5065:
5057:
5053:
5044:
5039:
5035:
5031:
5026:
5013:
5008:
5004:
4997:
4989:
4985:
4981:
4978:
4975:
4972:
4966:
4958:
4954:
4950:
4947:
4944:
4938:
4930:
4926:
4918:is given by:
4917:
4912:
4908:
4905:
4901:
4891:
4878:
4873:
4869:
4857:
4853:
4850:
4847:
4835:
4827:
4807:
4803:
4782:
4779:
4776:
4767:
4765:
4747:
4743:
4719:
4711:
4707:
4703:
4700:
4692:
4688:
4679:
4675:
4651:
4648:
4638:
4589:
4584:
4564:
4560:
4551:
4535:
4530:
4527:
4524:
4516:
4513:
4510:
4504:
4499:
4496:
4493:
4482:
4472:
4445:
4442:
4434:
4421:
4415:
4399:
4389:
4371:
4351:
4348:
4343:
4339:
4335:
4330:
4326:
4322:
4319:
4316:
4313:
4289:
4281:
4278:
4275:
4269:
4261:
4248:
4239:
4236:
4223:
4217:
4214:
4198:
4195:
4192:
4186:
4183:
4177:
4174:
4158:
4155:
4152:
4142:
4137:
4122:
4097:
4094:
4091:
4085:
4073:
4070:
4067:
4061:
4058:
4025:
4022:
4016:
4013:
3997:
3994:
3987:
3969:
3954:
3946:
3942:
3939:at the point
3924:
3904:
3886:
3883:
3880:
3866:be a line in
3865:
3861:
3844:
3840:
3831:
3818:
3815:
3812:
3809:
3805:
3802:
3794:
3777:
3773:
3764:
3760:
3748:
3735:
3732:
3717:
3714:
3711:
3706:
3702:
3693:
3689:
3680:
3676:
3672:
3661:
3647:
3624:
3599:
3595:
3590:
3584:
3580:
3573:
3551:
3520:
3506:
3482:
3466:
3463:
3460:
3456:
3449:
3444:
3440:
3433:
3429:
3425:
3420:
3416:
3412:
3407:
3403:
3399:
3394:
3390:
3383:
3375:
3371:
3366:
3360:
3356:
3349:
3344:
3340:
3315:
3312:
3307:
3303:
3291:
3278:
3275:
3253:
3232:
3221:
3218:
3215:
3205:
3202:
3182:
3177:
3174:
3171:
3142:
3138:
3134:
3131:
3128:
3123:
3119:
3110:
3109:
3107:
3104:
3102:
3101:tangent sheaf
3098:
3094:
3075:
3065:
3022:
2983:
2975:
2962:
2951:
2948:
2945:
2939:
2931:
2921:
2892:
2867:
2853:
2837:
2824:
2815:
2813:
2809:
2805:
2789:
2786:
2778:
2768:
2758:
2750:
2747:
2733:
2727:
2724:
2721:
2718:
2712:
2707:
2704:
2699:
2694:
2690:
2686:
2677:
2664:
2660:
2653:
2650:
2644:
2641:
2637:
2631:
2627:
2622:
2618:
2608:
2595:
2587:
2577:
2567:
2559:
2556:
2542:
2536:
2533:
2530:
2527:
2524:
2518:
2506:
2493:
2485:
2475:
2465:
2457:
2454:
2440:
2434:
2431:
2428:
2422:
2419:
2412:
2411:KĂ€hler metric
2409:Consider the
2407:
2394:
2391:
2385:
2377:
2373:
2363:
2350:
2347:
2336:
2331:
2313:
2309:
2299:
2297:
2293:
2292:
2287:
2283:
2267:
2260:
2253:
2231:
2218:
2215:
2212:
2204:
2201:
2197:
2193:
2189:
2171:
2142:
2140:
2135:
2132:
2112:
2109:
2106:
2101:
2094:
2091:
2088:
2072:
2065:
2058:
2055:
2052:
2043:
2040:
2031:
2027:
2021:
2018:
2011:
2005:
2002:
1991:
1983:
1978:
1972:
1964:
1960:
1939:
1936:
1933:
1930:
1927:
1924:
1902:
1896:
1893:
1882:
1862:
1859:
1845:
1841:
1838:
1831:
1826:
1813:
1807:
1804:
1793:
1776:
1773:
1767:
1762:
1751:
1731:
1724:
1721:
1717:
1711:
1708:
1697:
1689:
1685:
1668:
1665:
1656:
1652:
1648:
1644:
1640:
1636:
1632:
1628:
1611:
1608:
1600:
1597:
1576:
1570:
1567:
1563:
1560:
1551:
1547:
1543:
1540:
1524:
1518:
1515:
1512:
1503:
1477:
1472:
1468:
1459:
1454:
1452:
1442:
1429:
1425:
1421:
1418:
1413:
1409:
1400:
1396:
1392:
1385:
1363:
1338:
1319:
1316:
1308:
1289:
1286:
1280:
1277:
1272:
1268:
1259:
1255:
1251:
1244:
1222:
1214:
1192:
1189:
1183:
1180:
1155:
1152:
1149:
1145:
1141:
1136:
1132:
1125:
1117:
1113:
1107:
1103:
1094:
1077:
1074:
1071:
1065:
1062:
1054:
1029:
1023:
1020:
1003:
1000:
997:
991:
959:
947:
944:
941:
932:
926:
923:
903:
901:
897:
893:
889:
885:
881:
876:
874:
871:
867:
863:
859:
855:
854:indeterminate
851:
847:
843:
839:
835:
831:
827:
808:
805:
802:
794:
791:
786:
783:
780:
777:
766:
743:
739:
735:
719:
716:
713:
704:
689:
685:
678:
670:
666:
660:
656:
652:
648:
644:
641:
635:
632:
624:
621:
614:
602:
600:
577:
573:
569:
550:
542:
538:
529:
525:
522:
518:
515:
511:
508:
507:vector bundle
505:
499:
487:Constructions
484:
481:
479:
475:
471:
466:
464:
448:
441:
438:
429:
425:
421:
415:
407:
403:
399:
392:
389:
385:
382:
374:
370:
341:
338:
330:
326:
317:
313:
309:
304:
302:
298:
294:
290:
285:
283:
279:
278:homotopy type
275:
268:
258:
256:
251:
248:
244:
240:
235:
233:
229:
224:
222:
217:
215:
209:
207:
203:
199:
195:
191:
187:
183:
179:
174:
172:
168:
164:
160:
156:
152:
148:
144:
140:
137:(an infinite
136:
132:
127:
119:
117:
112:
110:
106:
102:
97:
94:
90:
75:
73:
69:
65:
61:
57:
53:
52:string theory
49:
46:
42:
38:
37:Chern classes
34:
30:
26:
22:
16206:Publications
16071:Chern number
16070:
16061:Betti number
15944: /
15935:Key concepts
15883:Differential
15775:
15757:
15731:
15704:
15701:Jost, JĂŒrgen
15681:
15646:
15642:
15615:
15582:
15576:
15555:
15551:
15089:
15085:
15076:
15013:
15001:
14989:
14977:
14965:
14957:
14953:
14795:
14783:
14770:
14761:
14740:
14721:
14712:
14653:
14637:
14632:
14628:
14624:
14620:
14616:
14613:Chern number
14612:
14608:
14600:
14595:, then each
14592:
14584:
14580:
14576:
14568:
14566:
14555:
14540:
14536:
14532:
14524:
14504:
14488:
14482:
14474:
14327:Chern number
14326:
14250:
14238:
14233:
14229:
14225:
14221:
14217:
14213:
14203:homomorphism
14198:
14196:
14048:
14045:
13935:
13930:
13926:
13920:
13654:
13651:
13373:
13239:
13234:
13228:
12575:is a degree
12539:
12322:
12009:
11807:
11625:
11468:
10890:
10870:
10866:
10826:
10819:
10815:
10769:
10765:
10759:
10751:
10520:
10518:
10513:
10509:
10505:
10356:The classes
10355:
10209:
10179:
10175:
10171:
10167:
10166:of the rank
10134:
10127:
10016:
9869:
9795:
9791:
9426:
9418:
9375:
9328:
9320:
9213:
9205:
9151:
9144:
9140:
9134:
9133:
9129:
9121:
9117:
9110:
9105:
8851:
7515:
7277:
7273:
7271:
7097:
7093:
6975:
6971:
6967:
6929:
6928:
6742:
6738:
6733:
6727:
6723:
6669:
6665:
6661:
6653:
6644:
6640:
6636:
6635:
6454:
6446:
6445:
6255:
6251:
5806:
5802:
5798:
5797:
5792:
5787:
5783:
5778:
5772:
5767:
5763:
5759:
5752:
5707:
5703:
5698:
5694:
5685:
5679:
5675:
5670:
5666:
5657:
5550:
5546:
5182:
5178:
5176:
5086:
5082:
5042:
5037:
5033:
5029:
5027:
4915:
4910:
4906:
4903:
4899:
4897:
4768:
4763:
4549:
4305:
3944:
3940:
3863:
3332:
3105:
2859:
2816:
2678:
2609:
2507:
2408:
2364:
2300:
2295:
2289:
2281:
2195:
2153:
2136:
2133:
1875:we see that
1683:
1654:
1650:
1646:
1642:
1641:(a point of
1638:
1634:
1630:
1626:
1549:
1545:
1541:
1504:
1455:
1448:
1095:
904:
899:
879:
877:
869:
865:
864:denotes the
861:
849:
845:
829:
764:
737:
733:
705:
603:
598:
567:
527:
523:
516:
514:complex rank
509:
501:
482:
467:
315:
305:
296:
288:
286:
273:
271:
252:
246:
236:
225:
218:
210:
205:
201:
197:
193:
189:
185:
181:
177:
175:
166:
162:
158:
154:
150:
146:
142:
139:Grassmannian
128:
125:
122:Construction
113:
98:
86:
36:
18:
16169:Wikiversity
16086:Key results
15649:: 137â154,
14718:Bott, Raoul
14683:Segre class
14678:Euler class
14545:cap product
14529:Chow groups
12372:Euler class
10875:Euler class
10122:Euler class
10021:line bundle
7379:dual bundle
6074:). The sum
5766:by writing
5547:Chern roots
3095:(i.e., the
2200:holomorphic
1686:. From the
842:gauge group
838:gauge field
824:with Ï the
476:classes of
301:Euler class
293:line bundle
91:. They are
60:knot theory
21:mathematics
16221:Categories
16015:CW complex
15956:Continuity
15946:Closed set
15905:cohomology
15566:References
15080:(See also
14788:Hartshorne
14640:symplectic
14450:partitions
14232:), and so
12374:. This is
10979:such that
9804:hyperplane
9541:direct sum
9415:continuous
9126:cohomology
9102:Properties
6451:Todd class
3986:hom bundle
2812:exact form
2137:See also:
528:Chern form
282:CW complex
16194:geometric
16189:algebraic
16040:Cobordism
15976:Hausdorff
15971:connected
15888:Geometric
15878:Continuum
15868:Algebraic
15665:0037-9484
15599:0003-486X
15514:…
15468:∑
15415:∏
15365:∏
15338:⊕
15335:⋯
15332:⊕
15234:∑
15179:…
15053:α
14909:↦
14869:∗
14855:η
14850:∞
14841:⊕
14837:→
14815:∗
14558:cobordism
14209:from the
14171:
14156:
14141:⊗
14132:
14101:
14083:
14068:⊕
14059:
14041:curvature
14006:π
13998:Ω
13985:
13974:
13951:
13904:⋯
13829:−
13759:−
13683:
13665:
13614:⋯
13573:∞
13558:∑
13531:⋯
13496:
13404:⊕
13401:⋯
13398:⊕
13315:∞
13300:∑
13268:
13250:
13174:−
13089:−
13062:−
12993:−
12687:→
12661:→
12623:→
12606:→
12551:⊂
12466:−
12447:−
12432:∫
12383:∫
12292:−
12242:−
12217:−
11841:∙
11791:→
11765:→
11727:→
11710:→
11604:→
11570:→
11532:→
11515:→
11480:⊂
11438:∙
11430:→
11373:↦
11199:→
11188:→
11182:→
11171:→
11066:(so that
10948:∈
10906:→
10893:Chow ring
10829:(meaning
10710:⋅
10693:−
10679:⋯
10668:−
10657:⋅
10622:−
10592:−
10553:…
10470:∗
10419:∗
10411:∈
10403:−
10389:…
10296:−
10287:τ
10254:τ
10193:∈
9983:⌣
9924:→
9913:→
9907:→
9896:→
9834:⊆
9826:−
9723:−
9712:⌣
9673:∑
9660:⊕
9636:that is,
9609:⌣
9585:⊕
9553:⊕
9524:→
9474:∗
9455:∗
9398:→
9304:⋯
8992:∈
8918:≅
8913:∗
8902:−
8690:
8532:
8379:∧
8316:α
8309:⋯
8290:α
8259:⋯
8242:∏
8219:∧
8173:α
8166:⋯
8147:α
8119:≤
8116:⋯
8113:≤
8099:∏
8082:
8009:α
8002:…
7990:α
7924:−
7896:∑
7889:…
7843:−
7811:−
7779:∑
7772:…
7699:…
7669:⊗
7618:−
7555:∑
7542:⊗
7467:−
7453:∗
7393:∗
7311:∑
7241:σ
7234:…
7222:σ
7181:∗
7156:−
7148:∗
7121:−
7112:
7062:−
7054:∗
7013:−
6903:σ
6894:∗
6820:∗
6812:→
6800:σ
6793:…
6781:σ
6764:∗
6696:→
6619:⋯
6522:−
6514:−
6484:∏
6468:
6429:⋯
6381:−
6331:−
6269:
6202:…
6164:∑
6129:⋯
6085:
6052:σ
6045:−
6031:σ
6005:σ
5952:…
5927:σ
5920:…
5898:…
5873:σ
5835:⋯
5618:…
5584:σ
5398:⋯
5312:⊕
5309:⋯
5306:⊕
5210:⊕
5140:⋯
4979:⋯
4780:≥
4748:∗
4704:−
4693:∗
4649:−
4352:⋯
4270:⊕
4237:−
4215:−
4199:
4193:⊕
4187:η
4175:−
4159:
4143:⊕
4086:⊕
4071:η
4068:⊕
4059:−
4026:η
4014:−
3998:
3813:≥
3800:→
3770:↦
3756:→
3733:−
3703:⊕
3686:↦
3669:→
3632:Ω
3628:→
3491:Ω
3464:≥
3413:−
3345:∗
3341:π
3297:∖
3239:→
3227:∖
3206::
3203:π
3132:…
3103:/bundle.
2981:→
2960:→
2940:⊕
2900:→
2871:→
2737:¯
2725:∧
2713:∫
2708:π
2687:∫
2657:Ω
2654:
2645:π
2546:¯
2534:∧
2516:Ω
2444:¯
2337:×
2265:∂
2257:∂
2019:−
2012:∗
1992:π
1937:−
1903:∗
1883:π
1860:⋯
1857:→
1832:
1814:∗
1794:π
1790:→
1768:
1755:→
1752:⋯
1729:→
1718::
1698:π
1605:→
1574:∖
1522:→
1516::
1513:π
1478:
1422:⋯
1397:π
1378:Ω
1364:−
1358:Ω
1335:Ω
1305:Ω
1287:−
1256:π
1237:Ω
1223:−
1211:Ω
1184:π
1173:Ω
1104:∑
1066:
1024:
1004:
948:
927:
809:ω
803:ω
784:ω
775:Ω
751:Ω
717:×
657:∑
636:π
628:Ω
585:Ω
504:hermitian
386:⊗
312:bijection
16159:Wikibook
16137:Category
16025:Manifold
15993:Homotopy
15951:Interior
15942:Open set
15900:Homology
15849:Topology
15756:(1974),
15729:(1999),
15703:(2005),
15637:(1958),
14662:See also
14597:monomial
14587:is also
14539:) to CH(
14211:K-theory
11362:The map
11264:″
11247:′
11195:″
11178:′
10865:, where
10814:for all
9996:″
9976:′
9920:″
9903:′
9794:, where
9374:for all
9106:Given a
8946:showing
7377:For the
6664:of rank
5282:Now, if
5263:′
5217:′
2392:≠
2145:Examples
2044:′
2006:′
1897:′
1842:′
1808:′
1725:′
1712:′
1669:′
1657:.) Then
1612:′
1601:′
1564:′
858:generate
840:for the
478:divisors
442:′
393:′
107:and the
16184:general
15986:uniform
15966:compact
15917:Digital
15673:0116023
15607:1969037
15096:in the
14589:compact
14039:is the
10772:, then
10512:, ...,
10120:is the
10023:, then
9874: (
9802:to the
9511:Whitney
9429:, then
9421:is the
9210:integer
9113:over a
7645:and so
7373:of it.
6746:reads:
6648:be the
6447:Example
5799:Example
5683:"are" Ï
5655:where Ï
4667:(note:
2186:be the
740:. The
574:of the
519:over a
239:section
70: (
45:complex
16179:Topics
15981:metric
15856:Fields
15782:
15764:
15739:
15715:
15689:
15671:
15663:
15623:
15605:
15597:
15006:Fulton
14994:Fulton
14982:Fulton
14970:Fulton
14728:
14398:, and
14035:where
10082:where
9939:is an
9132:. The
6930:Remark
6637:Remark
6449:: The
5444:where
4902:, the
4548:where
3832:where
2996:where
1952:. Let
35:, the
15961:Space
15603:JSTOR
14753:(PDF)
14704:Notes
14654:(See
14619:. If
14201:is a
13927:ch(V)
13212:is a
11086:is a
10818:>
10019:is a
9790:is 1â
9758:over
9208:with
8839:(cf.
8640:when
8482:when
6974:over
6668:over
5983:with
4735:when
2810:, an
2806:. By
2284:is a
2198:is a
978:that
884:up to
291:is a
280:of a
272:(Let
184:from
153:from
15780:ISBN
15762:ISBN
15737:ISBN
15713:ISBN
15687:ISBN
15661:ISSN
15621:ISBN
15595:ISSN
14726:ISBN
11808:Let
11338:rank
11334:>
11323:for
9876:1958
9417:and
8262:<
8256:<
7717:are
7369:the
7280:and
7272:Let
6991:Vect
6941:Vect
5661:are
3862:Let
3195:let
3111:Let
2154:Let
2110:>
2056:<
1928:<
1055:for
832:the
828:and
461:the
253:See
247:real
72:1946
39:are
31:and
15651:doi
15587:doi
15071:'s.
14615:of
14607:of
14579:of
14567:If
14516:or
14452:of
14205:of
13982:exp
13265:exp
12540:If
12469:200
12245:125
11630:in
11090:),
10764:If
9798:is
9425:of
9419:f*E
9413:is
9139:of
9128:of
8681:Sym
8523:Sym
8073:Sym
8027:of
7514:If
7407:of
7109:Nat
6652:of
6453:of
5795:)."
5782:by
5549:of
5085:of
4914:of
4766:.)
4196:Hom
4156:Hom
3995:Hom
3091:is
2294::
1649:of
1637:in
1629:of
1001:exp
986:det
954:det
856:to
844:of
763:of
611:det
597:of
566:of
512:of
472:by
284:.)
196:or
74:).
19:In
16223::
15811:,
15752:;
15711:,
15669:MR
15667:,
15659:,
15647:86
15645:,
15641:,
15601:,
15593:,
15583:47
15230::=
15123:,
14960:|.
14748:.
14658:)
14635:.
14564:.
14497:.
14472:.
14361:,
14243:.
14234:ch
14199:ch
14168:ch
14153:ch
14129:ch
14098:ch
14080:ch
14056:ch
13971:tr
13948:ch
13933:.
13680:rk
13662:ch
13554::=
13493:ch
13296::=
13247:ch
13216:.
12450:40
12295:40
12279:10
12229:25
12191:10
12175:10
12115:10
12099:10
11986:10
11970:10
10508:,
10353:.
10293:=:
10174:â
8843:.)
8782:11
8666:,
8508:,
8479:,
8047:,
7427:,
7101::
6577:12
6465:td
6284:rk
6266:ch
6082:ch
5089::
4795:,
3467:1.
2651:tr
2395:0.
2351:0.
2141:.
1744::
1643:BâČ
1469:GL
1453:.
1393:48
1063:ln
1021:ln
945:ln
924:ln
902:.
875:.
868:Ă
601:.
530:)
480:.
216:.
180:,
173:.
118:.
111:.
62:,
58:,
54:,
27:,
15841:e
15834:t
15827:v
15695:.
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15629:.
15589::
15556:V
15552:V
15530:)
15525:n
15521:x
15517:,
15511:,
15506:1
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15475:=
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15411:=
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15319:(
15316:c
15313:=
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15303:V
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15224:V
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15086:V
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15030:i
15026:c
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14938:2
14934:/
14929:|
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14900:]
14897:t
14894:[
14891:)
14888:)
14884:Z
14880:,
14877:M
14874:(
14866:2
14862:H
14858:(
14845:k
14834:)
14830:Z
14826:,
14823:M
14820:(
14812:2
14808:H
14778:.
14734:.
14633:M
14629:M
14625:M
14621:M
14617:M
14609:M
14601:d
14593:d
14585:M
14581:M
14569:M
14547:.
14541:V
14537:V
14533:V
14525:V
14520:.
14460:n
14436:n
14433:2
14411:3
14407:c
14384:2
14380:c
14374:1
14370:c
14347:3
14342:1
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14309:n
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14286:2
14266:n
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14230:X
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14218:X
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14214:K
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14177:W
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14144:W
14138:V
14135:(
14110:)
14107:W
14104:(
14095:+
14092:)
14089:V
14086:(
14077:=
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14071:W
14065:V
14062:(
14037:Ω
14022:]
14017:)
14012:)
14003:2
13995:i
13989:(
13978:(
13967:[
13963:=
13960:)
13957:V
13954:(
13931:V
13907:.
13901:+
13898:)
13895:)
13892:V
13889:(
13884:3
13880:c
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13870:)
13867:V
13864:(
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13855:c
13851:)
13848:V
13845:(
13840:1
13836:c
13832:3
13824:3
13820:)
13816:V
13813:(
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13804:c
13800:(
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13784:)
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13778:V
13775:(
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13766:c
13762:2
13754:2
13750:)
13746:V
13743:(
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13730:(
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13722:1
13717:+
13714:)
13711:V
13708:(
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13695:+
13692:)
13689:V
13686:(
13677:=
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13671:V
13668:(
13638:.
13635:)
13630:m
13625:n
13621:x
13617:+
13611:+
13606:m
13601:1
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13587:!
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13568:0
13565:=
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13470:)
13465:i
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13360:.
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13351:m
13344:m
13340:)
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13310:0
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13284:(
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13271:(
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13256:L
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13002:H
12999:[
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12987:(
12984:)
12979:2
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12971:H
12968:[
12965:6
12962:+
12959:]
12956:H
12953:[
12950:4
12947:+
12944:1
12941:(
12938:=
12925:)
12922:]
12919:H
12916:[
12913:d
12910:+
12907:1
12904:(
12897:4
12893:)
12889:]
12886:H
12883:[
12880:+
12877:1
12874:(
12868:=
12861:)
12856:X
12850:T
12844:(
12841:c
12814:)
12811:)
12808:d
12805:(
12800:X
12794:O
12788:(
12785:c
12780:)
12773:X
12768:|
12761:3
12756:P
12748:T
12742:(
12739:c
12733:=
12730:)
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12719:T
12713:(
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12681:d
12678:(
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12490:h
12463:=
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12439:X
12436:[
12428:=
12425:)
12420:X
12414:T
12408:(
12403:3
12399:c
12393:]
12390:X
12387:[
12358:)
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12341:(
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12332:c
12303:3
12299:h
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12283:h
12276:+
12273:1
12270:=
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12253:3
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12096:+
12093:h
12090:5
12087:+
12084:1
12078:=
12071:)
12066:X
12060:T
12054:(
12051:c
12025:4
12020:P
11994:3
11990:h
11983:+
11978:2
11974:h
11967:+
11964:h
11961:5
11958:+
11955:1
11952:=
11947:5
11943:)
11939:h
11936:+
11933:1
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11927:=
11924:)
11921:)
11918:5
11915:(
11910:X
11904:O
11898:(
11895:c
11892:)
11887:X
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11875:(
11872:c
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11849:X
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11837:A
11816:h
11794:0
11788:)
11785:5
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11760:X
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11741:P
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11676:X
11670:O
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11640:P
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11592:P
11586:/
11582:X
11576:N
11565:X
11560:|
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11538:T
11527:X
11521:T
11512:0
11490:n
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11434:A
11427:)
11424:X
11421:(
11416:0
11412:K
11408::
11405:c
11385:)
11382:E
11379:(
11376:c
11370:E
11348:)
11345:E
11342:(
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11299:(
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11268:)
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11257:(
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11244:E
11240:(
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11225:(
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11168:0
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7332:c
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7298:(
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7253:.
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7218:[
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7210:=
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7203:Z
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7194:n
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7118:[
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7076:.
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7059:(
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7029:]
7024:n
7020:G
7016:,
7010:[
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6995:n
6976:X
6972:n
6968:X
6951:C
6945:n
6915:.
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6907:k
6899:(
6889:E
6885:f
6881:=
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6875:E
6872:(
6867:k
6863:c
6842:.
6839:)
6835:Z
6831:,
6828:X
6825:(
6816:H
6809:]
6804:n
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6785:1
6777:[
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6769::
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6755:f
6743:E
6739:f
6734:k
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6693:X
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6681:f
6670:X
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6622:.
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6595:2
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6586:c
6582:(
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6564:1
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6546:+
6543:1
6540:=
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6526:a
6518:e
6511:1
6505:i
6501:a
6493:n
6488:1
6480:=
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6474:E
6471:(
6455:E
6432:.
6426:+
6423:)
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6317:c
6313:(
6308:2
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6300:+
6295:1
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6281:=
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6272:(
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6252:E
6238:!
6235:k
6231:/
6227:)
6224:)
6221:E
6218:(
6213:n
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6196:)
6193:E
6190:(
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6177:(
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6161:=
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6153:E
6150:(
6145:n
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6136:e
6132:+
6126:+
6121:)
6118:E
6115:(
6110:1
6106:a
6101:e
6097:=
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6091:E
6088:(
6056:2
6048:2
6040:2
6035:1
6027:=
6022:2
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6014:,
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5996:1
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5971:)
5968:)
5963:n
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5949:,
5944:1
5940:t
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5869:(
5864:k
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5856:=
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5846:n
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5838:+
5832:+
5827:k
5822:1
5818:t
5807:k
5803:s
5793:E
5791:(
5788:k
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5773:k
5768:f
5764:E
5760:f
5753:E
5739:)
5736:E
5733:(
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5699:i
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5680:k
5676:c
5671:i
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5643:)
5640:)
5637:E
5634:(
5629:n
5625:a
5621:,
5615:,
5612:)
5609:E
5606:(
5601:1
5597:a
5593:(
5588:k
5580:=
5577:)
5574:E
5571:(
5566:k
5562:c
5551:E
5533:)
5530:E
5527:(
5522:i
5518:a
5497:)
5492:i
5488:L
5484:(
5479:1
5475:c
5471:=
5468:)
5465:E
5462:(
5457:i
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5432:)
5429:t
5426:)
5423:E
5420:(
5415:n
5411:a
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5404:1
5401:(
5395:)
5392:t
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5386:E
5383:(
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5370:+
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5364:(
5361:=
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5355:E
5352:(
5347:t
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5320:n
5316:L
5301:1
5297:L
5293:=
5290:E
5270:.
5267:)
5260:E
5256:(
5251:t
5247:c
5243:)
5240:E
5237:(
5232:t
5228:c
5224:=
5221:)
5214:E
5207:E
5204:(
5199:t
5195:c
5183:t
5179:c
5162:)
5159:E
5156:(
5151:n
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5143:+
5137:+
5134:)
5131:E
5128:(
5123:1
5119:c
5115:+
5112:1
5109:=
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5103:E
5100:(
5097:c
5087:E
5069:)
5066:E
5063:(
5058:t
5054:c
5043:E
5041:(
5038:k
5034:c
5030:t
5014:.
5009:n
5005:t
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4998:E
4995:(
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4986:c
4982:+
4976:+
4973:t
4970:)
4967:E
4964:(
4959:1
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4951:+
4948:1
4945:=
4942:)
4939:E
4936:(
4931:t
4927:c
4916:E
4911:t
4907:c
4900:E
4879:.
4874:k
4870:a
4863:)
4858:k
4854:1
4851:+
4848:n
4842:(
4836:=
4833:)
4828:n
4823:P
4817:C
4813:(
4808:k
4804:c
4783:0
4777:k
4764:E
4744:E
4723:)
4720:E
4717:(
4712:1
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4701:=
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4689:E
4685:(
4680:1
4676:c
4655:)
4652:1
4646:(
4639:n
4634:P
4628:C
4621:O
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4594:Z
4590:,
4585:n
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4574:C
4570:(
4565:2
4561:H
4550:a
4536:,
4531:1
4528:+
4525:n
4521:)
4517:a
4514:+
4511:1
4508:(
4505:=
4500:1
4497:+
4494:n
4490:)
4486:)
4483:1
4480:(
4473:n
4468:P
4462:C
4455:O
4449:(
4446:c
4443:=
4440:)
4435:n
4430:P
4427:C
4422:T
4419:(
4416:c
4410:f
4407:e
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4400:=
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4390:n
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4379:C
4375:(
4372:c
4349:+
4344:2
4340:c
4336:+
4331:1
4327:c
4323:+
4320:1
4317:=
4314:c
4290:.
4285:)
4282:1
4279:+
4276:n
4273:(
4266:)
4262:1
4259:(
4254:O
4249:=
4246:)
4243:)
4240:1
4234:(
4229:O
4224:,
4221:)
4218:1
4212:(
4207:O
4202:(
4190:)
4184:,
4181:)
4178:1
4172:(
4167:O
4162:(
4153:=
4148:O
4138:n
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4127:C
4123:T
4101:)
4098:1
4095:+
4092:n
4089:(
4080:O
4074:=
4065:)
4062:1
4056:(
4051:O
4029:)
4023:,
4020:)
4017:1
4011:(
4006:O
4001:(
3970:n
3965:P
3959:C
3955:T
3945:L
3941:L
3925:n
3920:P
3914:C
3887:1
3884:+
3881:n
3876:C
3864:L
3845:i
3841:e
3819:,
3816:0
3810:i
3806:,
3803:0
3795:U
3789:O
3778:i
3774:z
3765:i
3761:e
3749:U
3744:|
3739:)
3736:1
3730:(
3725:O
3718:1
3715:+
3712:n
3707:1
3694:i
3690:e
3681:i
3677:z
3673:d
3662:U
3657:|
3648:n
3643:P
3637:C
3625:0
3605:)
3600:0
3596:z
3591:/
3585:i
3581:z
3577:(
3574:d
3552:U
3546:O
3521:U
3516:|
3507:n
3502:P
3496:C
3461:i
3457:,
3450:2
3445:0
3441:z
3434:0
3430:z
3426:d
3421:i
3417:z
3408:i
3404:z
3400:d
3395:0
3391:z
3384:=
3381:)
3376:0
3372:z
3367:/
3361:i
3357:z
3353:(
3350:d
3319:}
3316:0
3313:=
3308:0
3304:z
3300:{
3292:n
3287:P
3284:C
3279:=
3276:U
3254:n
3249:P
3243:C
3236:}
3233:0
3230:{
3222:1
3219:+
3216:n
3211:C
3183:,
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3172:n
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3143:n
3139:z
3135:,
3129:,
3124:0
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3079:)
3076:1
3073:(
3066:n
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3058:C
3050:O
3023:n
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3007:O
2984:0
2976:n
2971:P
2968:C
2963:T
2955:)
2952:1
2949:+
2946:n
2943:(
2936:)
2932:1
2929:(
2922:n
2917:P
2914:C
2906:O
2893:n
2888:P
2885:C
2877:O
2868:0
2838:1
2833:P
2830:C
2825:T
2790:2
2787:=
2779:2
2775:)
2769:2
2764:|
2759:z
2755:|
2751:+
2748:1
2745:(
2734:z
2728:d
2722:z
2719:d
2705:i
2700:=
2695:1
2691:c
2665:.
2661:]
2642:2
2638:i
2632:[
2628:=
2623:1
2619:c
2596:.
2588:2
2584:)
2578:2
2573:|
2568:z
2564:|
2560:+
2557:1
2554:(
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2519:=
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2466:z
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2458:+
2455:1
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2429:d
2423:=
2420:h
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2386:V
2383:(
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2348:=
2345:)
2341:C
2332:1
2327:P
2324:C
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2310:c
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2282:a
2268:z
2261:/
2254:a
2232:1
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2216:=
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2113:n
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2092:=
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2069:(
2066:e
2059:n
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2037:(
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2003:B
1997:|
1984:{
1979:=
1976:)
1973:E
1970:(
1965:k
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1934:n
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1771:(
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1732:B
1722:B
1709:B
1703:|
1684:E
1666:E
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1647:F
1639:F
1635:v
1631:E
1627:F
1609:B
1598:E
1577:B
1571:E
1568:=
1561:B
1550:E
1546:B
1542:B
1525:B
1519:E
1489:)
1485:C
1481:(
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1430:.
1426:]
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1355:(
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1331:(
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1301:(
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1234:(
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1190:t
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1146:[
1142:=
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1133:t
1129:)
1126:V
1123:(
1118:k
1114:c
1108:k
1081:)
1078:I
1075:+
1072:X
1069:(
1039:)
1036:)
1033:)
1030:X
1027:(
1018:(
1014:r
1011:t
1007:(
998:=
995:)
992:X
989:(
966:)
963:)
960:X
957:(
951:(
942:=
939:)
936:)
933:X
930:(
921:(
917:r
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870:n
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862:I
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830:d
812:]
806:,
800:[
795:2
792:1
787:+
781:d
778:=
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738:M
734:t
720:n
714:n
690:k
686:t
682:)
679:V
676:(
671:k
667:c
661:k
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645:I
642:+
633:2
625:t
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615:(
599:V
568:V
554:)
551:V
548:(
543:k
539:c
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517:n
510:V
449:;
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439:L
435:(
430:1
426:c
422:+
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416:L
413:(
408:1
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400:=
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390:L
383:L
380:(
375:1
371:c
350:)
346:Z
342:;
339:X
336:(
331:2
327:H
316:X
297:X
289:V
274:X
269:.
206:V
202:M
198:g
194:f
190:V
186:M
182:g
178:f
167:V
163:f
159:V
155:M
151:f
147:M
143:V
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