6557:, with relations that the sum and product of the ring on these generators are given by disjoint union and product of manifolds (with the obvious operations on the vector bundles), and any boundary of a manifold with vector bundle is 0. This is similar to the cobordism ring of oriented manifolds, except that the manifolds also have a vector bundle. The topological and analytical indices are both reinterpreted as functions from this ring to the integers. Then one checks that these two functions are in fact both ring homomorphisms. In order to prove they are the same, it is then only necessary to check they are the same on a set of generators of this ring. Thom's cobordism theory gives a set of generators; for example, complex vector spaces with the trivial bundle together with certain bundles over even dimensional spheres. So the index theorem can be proved by checking it on these particularly simple cases.
4899:
4221:
4894:{\displaystyle {\begin{aligned}\operatorname {ch} {\mathord {\left(\Lambda ^{\text{even}}-\Lambda ^{\text{odd}}\right)}}&=1-\operatorname {ch} (T^{*}M\otimes \mathbb {C} )+\operatorname {ch} {\mathord {\left(\Lambda ^{2}T^{*}M\otimes \mathbb {C} \right)}}-\ldots +(-1)^{n}\operatorname {ch} {\mathord {\left(\Lambda ^{n}T^{*}M\otimes \mathbb {C} \right)}}\\&=1-\sum _{i}^{n}e^{-x_{i}}(TM\otimes \mathbb {C} )+\sum _{i<j}e^{-x_{i}}e^{-x_{j}}(TM\otimes \mathbb {C} )+\ldots +(-1)^{n}e^{-x_{1}}\dotsm e^{-x_{n}}(TM\otimes \mathbb {C} )\\&=\prod _{i}^{n}\left(1-e^{-x_{i}}\right)(TM\otimes \mathbb {C} )\\\operatorname {Td} (TM\otimes \mathbb {C} )&=\prod _{i}^{n}{\frac {x_{i}}{1-e^{-x_{i}}}}(TM\otimes \mathbb {C} )\end{aligned}}}
6146:
5741:
2878:
6141:{\displaystyle {\begin{aligned}\operatorname {ch} \left(\sum _{j}^{n}(-1)^{j}V\otimes \Lambda ^{j}{\overline {T^{*}X}}\right)&=\operatorname {ch} (V)\prod _{j}^{n}\left(1-e^{x_{j}}\right)(TX)\\\operatorname {Td} (TX\otimes \mathbb {C} )=\operatorname {Td} (TX)\operatorname {Td} \left({\overline {TX}}\right)&=\prod _{i}^{n}{\frac {x_{i}}{1-e^{-x_{i}}}}\prod _{j}^{n}{\frac {-x_{j}}{1-e^{x_{j}}}}(TX)\end{aligned}}}
5251:
3214:
as an elliptic operator between two vector bundles. Conversely the index theorem for an elliptic complex can easily be reduced to the case of an elliptic operator: the two vector bundles are given by the sums of the even or odd terms of the complex, and the elliptic operator is the sum of the operators of the elliptic complex and their adjoints, restricted to the sum of the even bundles.
2655:
4910:
6845:
3213:
of vector bundles. The difference is that the symbols now form an exact sequence (off the zero section). In the case when there are just two non-zero bundles in the complex this implies that the symbol is an isomorphism off the zero section, so an elliptic complex with 2 terms is essentially the same
1103:
has index 0. This example shows that the kernel and cokernel of elliptic operators can jump discontinuously as the elliptic operator varies, so there is no nice formula for their dimensions in terms of continuous topological data. However the jumps in the dimensions of the kernel and cokernel are the
2585:
In spite of its formidable definition, the topological index is usually straightforward to evaluate explicitly. So this makes it possible to evaluate the analytical index. (The cokernel and kernel of an elliptic operator are in general extremely hard to evaluate individually; the index theorem shows
6501:
Many proofs of the index theorem use pseudodifferential operators rather than differential operators. The reason for this is that for many purposes there are not enough differential operators. For example, a pseudoinverse of an elliptic differential operator of positive order is not a differential
3221:
is allowed to have boundary, then some restrictions must be put on the domain of the elliptic operator in order to ensure a finite index. These conditions can be local (like demanding that the sections in the domain vanish at the boundary) or more complicated global conditions (like requiring that
2593:
Although the analytical index is usually hard to evaluate directly, it is at least obviously an integer. The topological index is by definition a rational number, but it is usually not at all obvious from the definition that it is also integral. So the Atiyah–Singer index theorem implies some deep
6517:
Pseudodifferential operators have an order, which can be any real number or even −∞, and have symbols (which are no longer polynomials on the cotangent space), and elliptic differential operators are those whose symbols are invertible for sufficiently large cotangent vectors. Most versions of the
6467:
is a rational number defined for any manifold, but is in general not an integer. Borel and
Hirzebruch showed that it is integral for spin manifolds, and an even integer if in addition the dimension is 4 mod 8. This can be deduced from the index theorem, which implies that the  genus for spin
2873:{\displaystyle {\begin{array}{ccc}&&&\\&K(X)&{\xrightarrow{{\text{Td}}(X)\cdot {\text{ch}}}}&H(X;\mathbb {Q} )&\\&f_{*}{\Bigg \downarrow }&&{\Bigg \downarrow }f_{*}\\&K(Y)&{\xrightarrow{}}&H(Y;\mathbb {Q} )&\\&&&\\\end{array}}}
3071:
It is important to mention that the index formula is a topological statement. The obstruction theories due to Milnor, Kervaire, Kirby, Siebenmann, Sullivan, Donaldson show that only a minority of topological manifolds possess differentiable structures and these are not necessarily unique.
3303:
showed how to extend the index theorem to some non-compact manifolds, acted on by a discrete group with compact quotient. The kernel of the elliptic operator is in general infinite dimensional in this case, but it is possible to get a finite index using the dimension of a module over a
6468:
manifolds is the index of a Dirac operator. The extra factor of 2 in dimensions 4 mod 8 comes from the fact that in this case the kernel and cokernel of the Dirac operator have a quaternionic structure, so as complex vector spaces they have even dimensions, so the index is even.
5489:
182:
The Atiyah–Singer theorem was announced in 1963. The proof sketched in this announcement was never published by them, though it appears in Palais's book. It appears also in the "Séminaire Cartan-Schwartz 1963/64" that was held in Paris simultaneously with the seminar led by
1354:
4090:
5246:{\displaystyle \chi (M)=(-1)^{r}\int _{M}{\frac {\prod _{i}^{n}\left(1-e^{-x_{i}}\right)}{\prod _{i}^{r}x_{i}}}\prod _{i}^{n}{\frac {x_{i}}{1-e^{-x_{i}}}}(TM\otimes \mathbb {C} )=(-1)^{r}\int _{M}(-1)^{r}\prod _{i}^{r}x_{i}(TM\otimes \mathbb {C} )=\int _{M}e(TM)}
913:); however, the highest order terms transform like tensors so we get well defined homogeneous functions on the cotangent spaces that are independent of the choice of local charts.) More generally, the symbol of a differential operator between two vector bundles
3211:
2410:
3892:
3238:
introduced global boundary conditions equivalent to attaching a cylinder to the manifold along the boundary and then restricting the domain to those sections that are square integrable along the cylinder. This point of view is adopted in the proof of
992:
is finite-dimensional, because all eigenspaces of compact operators, other than the kernel, are finite-dimensional. (The pseudoinverse of an elliptic differential operator is almost never a differential operator. However, it is an elliptic
3120:
in much the same way as for elliptic differential operators. In fact, for technical reasons most of the early proofs worked with pseudodifferential rather than differential operators: their extra flexibility made some steps of the proofs
6701:
6419:
6866:
appear very complicated, but invariant theory shows that there are huge cancellations between the terms, which makes it possible to find the leading terms explicitly. These cancellations were later explained using supersymmetry.
5658:
1875:
6237:
1739:
1625:
2590:.) Many important invariants of a manifold (such as the signature) can be given as the index of suitable differential operators, so the index theorem allows us to evaluate these invariants in terms of topological data.
6682:
are self adjoint operators whose non-zero eigenvalues have the same multiplicities. However their zero eigenspaces may have different multiplicities, as these multiplicities are the dimensions of the kernels of
4226:
5356:
5374:
4213:
3244:
3132:
1407:
227:'s paper proved the existence of rational Pontryagin classes on topological manifolds. The rational Pontryagin classes are essential ingredients of the index theorem on smooth and topological manifolds.
5663:
Since we are dealing with complex bundles, the computation of the topological index is simpler. Using Chern roots and doing similar computations as in the previous example, the Euler class is given by
3965:
1222:
233:
Michael Atiyah defines abstract elliptic operators on arbitrary metric spaces. Abstract elliptic operators became protagonists in
Kasparov's theory and Connes's noncommutative differential geometry.
3970:
555:
5746:
3049:) "provides local constructions for characteristic classes based on higher dimensional relatives of the measurable Riemann mapping in dimension two and the Yang–Mills theory in dimension four."
8070:
Séminaire Henri Cartan. Théoreme d'Atiyah-Singer sur l'indice d'un opérateur différentiel elliptique. 16 annee: 1963/64 dirigee par Henri Cartan et
Laurent Schwartz. Fasc. 1; Fasc. 2. (French)
6494:
Pseudodifferential operators can be explained easily in the case of constant coefficient operators on
Euclidean space. In this case, constant coefficient differential operators are just the
3334:
is an index theorem for a Dirac operator on a noncompact odd-dimensional space. The Atiyah–Singer index is only defined on compact spaces, and vanishes when their dimension is odd. In 1978
962:
A key property of elliptic operators is that they are almost invertible; this is closely related to the fact that their symbols are almost invertible. More precisely, an elliptic operator
2298:
854:
1950:
768:
5733:
5368:
This derivation of the
Hirzebruch–Riemann–Roch theorem is more natural if we use the index theorem for elliptic complexes rather than elliptic operators. We can take the complex to be
3800:
670:
597:
5519:
3795:
2617:
theorem was one of the main motivations behind the index theorem because the index theorem is the counterpart of this theorem in the setting of real manifolds. Now, if there's a map
3580:
3487:
1524:
1466:
3607:
3514:
2489:
One can also define the topological index using only K-theory (and this alternative definition is compatible in a certain sense with the Chern-character construction above). If
469:
4130:
1801:
3222:
the sections in the domain solve some differential equation). The local case was worked out by Atiyah and Bott, but they showed that many interesting operators (e.g., the
2005:
6337:
2455:
2183:
2123:
3553:
3659:
2647:
880:
6498:
of multiplication by polynomials, and constant coefficient pseudodifferential operators are just the
Fourier transforms of multiplication by more general functions.
2065:
2035:
6840:{\displaystyle \operatorname {index} (D)=\dim \operatorname {Ker} (D^{*})=\operatorname {Tr} \left(e^{-tD^{*}D}\right)-\operatorname {Tr} \left(e^{-tDD^{*}}\right)}
2286:
1104:
same, so the index, given by the difference of their dimensions, does indeed vary continuously, and can be given in terms of topological data by the index theorem.
795:
697:
624:
3747:
3460:
2936:
2152:
2094:
2907:
2484:
6356:
3331:
2251:
3718:
3683:
3627:
3431:
2228:
2203:
1490:
1431:
1214:
1194:
1174:
1154:
1134:
1091:
is the operator d/dx − λ for some complex constant λ. (This is the simplest example of an elliptic operator.) Then the kernel is the space of multiples of exp(λ
2942:
of complex vector bundles. This commutative diagram is formally very similar to the GRR theorem because the cohomology groups on the right are replaced by the
6475:
that the signature of a 4-dimensional spin manifold is divisible by 16: this follows because in dimension 4 the  genus is minus one eighth of the signature.
3335:
9046:
5547:
1807:
6854:. The right hand side is given by the trace of the difference of the kernels of two heat operators. These have an asymptotic expansion for small positive
3378:
8666:
This describes the original proof of the theorem (Atiyah and Singer never published their original proof themselves, but only improved versions of it.)
6161:
1632:
8065:
1529:
9133:
3076:) shows that any topological manifold in dimension different from 4 possesses such a structure which is unique (up to isotopy close to identity).
3382:
2981:
The proof of this result goes through specific considerations, including the extension of Hodge theory on combinatorial and
Lipschitz manifolds (
3041:
Using topological cobordism and K-homology one may provide a full statement of an index theorem on quasiconformal manifolds (see page 678 of (
7592:
This paper studies families of elliptic operators, where the index is now an element of the K-theory of the space parametrizing the family.
3060:). At the same time, they provide, also, an effective construction of the rational Pontrjagin classes on topological manifolds. The paper (
5314:
7630:. This studies families of real (rather than complex) elliptic operators, when one can sometimes squeeze out a little extra information.
3354:. The index of the Dirac operator is a topological invariant which measures the winding of the Higgs field on a sphere at infinity. If
9138:
5484:{\displaystyle 0\rightarrow V\rightarrow V\otimes \Lambda ^{0,1}T^{*}(X)\rightarrow V\otimes \Lambda ^{0,2}T^{*}(X)\rightarrow \dotsm }
2614:
75:
2597:
The index of an elliptic differential operator obviously vanishes if the operator is self adjoint. It also vanishes if the manifold
909:
on each cotangent space. (In general, differential operators transform in a rather complicated way under coordinate transforms (see
4138:
699:
because we keep only the highest order terms and differential operators commute "up to lower-order terms". The operator is called
299:
Nicolae
Teleman proves that the analytical indices of signature operators with values in vector bundles are topological invariants.
9021:
6502:
operator, but is a pseudodifferential operator. Also, there is a direct correspondence between data representing elements of K(B(
1365:
1349:{\displaystyle (-1)^{n}\operatorname {ch} (D)\operatorname {Td} (X)=(-1)^{n}\int _{X}\operatorname {ch} (D)\operatorname {Td} (X)}
6527:
156:
4085:{\displaystyle x_{r+i}(E\otimes \mathbb {C} )=c_{1}{\mathord {\left({\overline {l_{i}}}\right)}}=-x_{i}(E\otimes \mathbb {C} )}
3897:
8570:, Annals of Mathematics Studies in Mathematics, vol. 88, Princeton: Princeton University Press and Tokio University Press
8730:
8697:
8660:
8611:
8390:
8143:
8029:
7952:
7923:
7248:
9115:- A partial transcript of informal post–dinner conversation during a symposium held in Roskilde, Denmark, in September 1998.
2946:
of a smooth variety, and the
Grothendieck group on the left is given by the Grothendieck group of algebraic vector bundles.
8207:; Teleman, N. (1994), "Quasiconformal mappings, operators on Hilbert space and local formulae for characteristic classes",
482:
8310:. On page 120 Gel'fand suggests that the index of an elliptic operator should be expressible in terms of topological data.
3004:
This result shows that the index theorem is not merely a differentiability statement, but rather a topological statement.
3124:
Instead of working with an elliptic operator between two vector bundles, it is sometimes more convenient to work with an
327:
and
Sullivan study Yang–Mills theory on quasiconformal manifolds of dimension 4. They introduce the signature operator
8981:
5256:
which is the "topological" version of the Chern-Gauss-Bonnet theorem (the geometric one being obtained by applying the
3690:
2571:
data derived from the manifold and the vector bundle. The Atiyah–Singer index theorem solves this problem, and states:
124:
71:
8400:
Hamilton, M. J. D. (2020). "The Higgs boson for mathematicians. Lecture notes on gauge theory and symmetry breaking".
8035:
This gives an elementary proof of the index theorem for the Dirac operator, using the heat equation and supersymmetry.
8591:
8439:
8307:
7977:
7349:
7279:
3206:{\displaystyle 0\rightarrow E_{0}\rightarrow E_{1}\rightarrow E_{2}\rightarrow \dotsm \rightarrow E_{m}\rightarrow 0}
2405:{\displaystyle (-1)^{m}\int _{X}{\frac {\operatorname {ch} (E)-\operatorname {ch} (F)}{e(TX)}}\operatorname {Td} (X)}
472:
6627:
of some elliptic operator on the point. This reduces the index theorem to the case of a point, where it is trivial.
3749:, in order to prove assertions involving characteristic classes, we may suppose that there are complex line bundles
5534:
3250:
Instead of just one elliptic operator, one can consider a family of elliptic operators parameterized by some space
1099:
and is 0 otherwise, and the kernel of the adjoint is a similar space with λ replaced by its complex conjugate. So
7294:
3887:{\displaystyle E\otimes \mathbb {C} =l_{1}\oplus {\overline {l_{1}}}\oplus \dotsm l_{r}\oplus {\overline {l_{r}}}}
800:
9128:
9112:
6152:
1885:
717:
8061:
Bismut proves the theorem for elliptic complexes using probabilistic methods, rather than heat equation methods.
3025:
For any quasiconformal manifold there exists a local construction of the Hirzebruch–Thom characteristic classes.
9077:
8621:
7973:
5666:
3320:
3282:
3258:, rather than an integer. If the operators in the family are real, then the index lies in the real K-theory of
2210:
In some situations, it is possible to simplify the above formula for computational purposes. In particular, if
206:
6518:
index theorem can be extended from elliptic differential operators to elliptic pseudodifferential operators.
637:
564:
175:
of a spin manifold, and Atiyah suggested that this integrality could be explained if it were the index of the
9066:
5497:
3752:
3286:
7698:
Atiyah, M. F.; Bott, R. (1968), "A Lefschetz Fixed Point Formula for Elliptic Complexes: II. Applications",
3558:
3465:
1497:
1439:
191:. The last talk in Paris was by Atiyah on manifolds with boundary. Their first published proof replaced the
6986:
6535:
6259:
3585:
3492:
160:
79:
17:
3358:
is the unit matrix in the direction of the Higgs field, then the index is proportional to the integral of
343:
Connes, Sullivan, and Teleman prove the index theorem for signature operators on quasiconformal manifolds.
9061:
8318:
6489:
3117:
994:
428:
8633:
4095:
3270:
1410:
245:
Gennadi G. Kasparov publishes his work on the realization of K-homology by abstract elliptic operators.
1748:
7969:
3294:
3034:, defined on middle degree differential forms on even-dimensional quasiconformal manifolds (compare (
9056:
8680:
7450:
This reformulates the result as a sort of Lefschetz fixed-point theorem, using equivariant K-theory.
7231:
6862:
tends to 0, giving a proof of the Atiyah–Singer index theorem. The asymptotic expansions for small
337:
Connes and Henri Moscovici prove the local index formula in the context of non-commutative geometry.
7660:. This states a theorem calculating the Lefschetz number of an endomorphism of an elliptic complex.
6242:
In fact we get a generalization of it to all complex manifolds: Hirzebruch's proof only worked for
152:
128:
1955:
8579:
7328:
7317:
7267:
5257:
2976:) on a closed, oriented, topological manifold, the analytical index equals the topological index.
315:
6300:
2418:
2159:
2099:
8675:
7226:
3519:
273:
6289:
are given by the +1 and −1 eigenspaces of the operator on the bundle of differential forms of
3635:
2620:
199:, and they used this to give proofs of various generalizations in another sequence of papers.
147:. He noticed the homotopy invariance of the index, and asked for a formula for it by means of
8563:
8534:
7700:
7605:
7567:
7529:
859:
220:
148:
92:
39:
6274:
of the manifold. This follows from the Atiyah–Singer index theorem applied to the following
2040:
2010:
8996:
8962:
8799:
8511:
8330:
8286:
7962:
7933:
7902:
7880:
7857:
7837:
7814:
7792:
7769:
7749:
7725:
These give the proofs and some applications of the results announced in the previous paper.
7411:
7366:
The papers by Atiyah are reprinted in volumes 3 and 4 of his collected works, (Atiyah
6472:
6414:{\displaystyle D\equiv \Delta \mathrel {:=} \left(\mathbf {d} +\mathbf {d^{*}} \right)^{2}}
3630:
3394:
3324:
3064:) provides a link between Thom's original construction of the rational Pontrjagin classes (
2525:
in Euclidean space. Now a differential operator as above naturally defines an element of K(
2259:
773:
675:
602:
188:
164:
123:(defined in terms of some topological data). It includes many other theorems, such as the
83:
8889:
8859:
8783:
8740:
8267:
8230:
8193:
8153:
8120:
8077:
8013:
7392:
6541:
The idea of this first proof is roughly as follows. Consider the ring generated by pairs (
3723:
3436:
2912:
2128:
2070:
8:
6433:
3697:
3305:
2886:
2660:
2463:
132:
9097:
8803:
8515:
8396:
Free online textbook that proves the Atiyah–Singer theorem with a heat equation approach
8334:
8290:
7884:
7841:
7796:
7753:
7554:
This paper shows how to convert from the K-theory version to a version using cohomology.
4135:
Using Chern roots as above and the standard properties of the Euler class, we have that
2233:
897:
is defined in much the same way using local coordinate charts, and is a function on the
396:. So in local coordinates it acts as a differential operator, taking smooth sections of
8913:
8847:
8815:
8771:
8723:
J.C. Candrell, "Geometric Topology", Proc. Georgia Topology Conf. Athens, Georgia, 1977
8491:
8483:
8465:
8416:
8401:
8346:
8132:
8108:
8039:
8001:
7947:, Oxford Science Publications, New York: The Clarendon Press, Oxford University Press,
7918:, Oxford Science Publications, New York: The Clarendon Press, Oxford University Press,
7861:
7773:
7737:
7717:
7688:
7622:
7584:
7546:
7508:
7442:
6640:
6275:
3703:
3668:
3612:
3416:
3308:; this index is in general real rather than integer valued. This version is called the
3231:
3223:
2939:
2213:
2188:
1475:
1416:
1199:
1179:
1159:
1139:
1119:
286:
144:
8502:
Kasparov, G.G. (1972), "Topological invariance of elliptic operators, I: K-homology",
8042:(1984), "The Atiyah–Singer Theorems: A Probabilistic Approach. I. The index theorem",
239:
Isadore Singer proposes a comprehensive program for future extensions of index theory.
9073:
9040:
8977:
8917:
8851:
8819:
8726:
8693:
8656:
8607:
8587:
8495:
8435:
8386:
8371:
8303:
8221:
8184:
8139:
8112:
8056:
8025:
7981:
7948:
7919:
7865:
7828:(1977), "A geometric construction of the discrete series for semisimple Lie groups",
7777:
7345:
7275:
7244:
6495:
5653:{\displaystyle \operatorname {index} (D)=\sum _{p}(-1)^{p}\dim H^{p}(X,V)=\chi (X,V)}
2458:
1870:{\displaystyle \operatorname {ch} :K(X)\otimes \mathbb {Q} \to H^{*}(X;\mathbb {Q} )}
1010:
108:
8827:
8751:
8650:
8551:
8523:
8350:
8319:"Pseudodifferential operators on supermanifolds and the Atiyah–Singer index theorem"
8298:
8088:
8005:
7655:
7475:
3386:
2521:(as a consequence of Bott-periodicity). This map is independent of the embedding of
9113:
Recollections from the early days of index theory and pseudo-differential operators
8948:
8905:
8885:
8875:
8855:
8839:
8807:
8779:
8775:
8763:
8736:
8685:
8575:
8546:
8519:
8475:
8366:
8338:
8294:
8263:
8253:
8226:
8216:
8189:
8179:
8149:
8116:
8100:
8073:
8051:
8009:
7993:
7888:
7845:
7800:
7757:
7709:
7680:
7650:
7614:
7576:
7538:
7500:
7470:
7434:
7388:
7263:
7236:
6531:
5273:
3662:
3347:
3126:
2289:
1742:
1360:
1018:
898:
210:
112:
3052:
These results constitute significant advances along the lines of Singer's program
8958:
8747:
8718:
8415:
Kayani, U. (2020). "Dynamical supersymmetry enhancement of black hole horizons".
8241:
8237:
8204:
7958:
7929:
7913:
7898:
7853:
7825:
7810:
7765:
7407:
6348:
6232:{\displaystyle \chi (X,V)=\int _{X}\operatorname {ch} (V)\operatorname {Td} (TX)}
3281:, commuting with the elliptic operator, then one replaces ordinary K-theory with
1878:
1734:{\displaystyle \varphi :H^{k}(X;\mathbb {Q} )\to H^{n+k}(B(X)/S(X);\mathbb {Q} )}
324:
289:, gave a short proof of the local index theorem for operators that are locally
269:
2649:
of compact stably almost complex manifolds, then there is a commutative diagram
9000:
8988:
8969:
8706:
8646:
8274:
7940:
7909:
7821:
7729:
7664:
7641:(1966), "A Lefschetz Fixed Point Formula for Elliptic Differential Operators",
7634:
7600:
7596:
7562:
7558:
7524:
7520:
7488:
7484:
7458:
7454:
7418:
7399:
7380:
6514:)) (clutching functions) and symbols of elliptic pseudodifferential operators.
3343:
3235:
3227:
3103:), are the weakest analytical structures on topological manifolds of dimension
2989:), the extension of Atiyah–Singer's signature operator to Lipschitz manifolds (
2769:
2759:
1620:{\displaystyle \varphi ^{-1}(\operatorname {ch} (d(p^{*}E,p^{*}F,\sigma (D))))}
290:
184:
176:
104:
100:
53:
49:
8927:(1956), "Les classes caractéristiques de Pontrjagin de variétés triangulées",
8713:, Annals of Mathematics Studies in Mathematics, vol. 70, pp. 171–185
8357:
Getzler, E. (1988), "A short proof of the local Atiyah–Singer index theorem",
8164:
3262:. This gives a little extra information, as the map from the real K-theory of
2594:
integrality properties, as it implies that the topological index is integral.
9122:
8953:
8936:
6648:
988:
are both compact operators. An important consequence is that the kernel of
375:
357:
260:
3696:
The concrete computation goes as follows: according to one variation of the
2292:
and dividing by the Euler class, the topological index may be expressed as
8655:, Annals of Mathematics Studies, vol. 57, S.l.: Princeton Univ Press,
8568:
Foundational Essays on Topological Manifolds, Smoothings and Triangulations
8559:
8530:
8314:
8200:
8160:
8127:
8084:
7422:
6885:
3339:
311:
282:
256:
216:
168:
8924:
7404:
Colloque "Analyse et Topologie" en l'Honneur de Henri Cartan (Orsay, 1974)
1037:(the constraints on the right-hand-side of an inhomogeneous equation like
224:
8790:
Teleman, N. (1980), "Combinatorial Hodge theory and signature operator",
8625:
8601:
7406:, Asterisque, vol. 32–33, Soc. Math. France, Paris, pp. 43–72,
3686:
3377:
The topological interpretation of this invariant and its relation to the
3351:
2254:
272:
establishes his theorem on the existence and uniqueness of Lipschitz and
8832:
Publications Mathématiques de l'Institut des Hautes Études Scientifiques
8756:
Publications Mathématiques de l'Institut des Hautes Études Scientifiques
8380:
8093:
Publications Mathématiques de l'Institut des Hautes Études Scientifiques
7402:(1976), "Elliptic operators, discrete groups and von Neumann algebras",
7385:
Proc. Int. Conf. on Functional Analysis and Related Topics (Tokyo, 1969)
8992:
8909:
8880:
8843:
8811:
8767:
8752:"An analytic proof of Novikov's theorem on rational Pontrjagin classes"
8689:
8342:
8258:
8104:
7997:
7893:
7849:
7805:
7761:
7733:
7721:
7692:
7668:
7638:
7626:
7588:
7550:
7512:
7446:
7240:
6636:
6340:
3489:
to be the sum of the even exterior powers of the cotangent bundle, and
3390:
1469:
1359:
in other words the value of the top dimensional component of the mixed
1045:, or equivalently the kernel of the adjoint operator). In other words,
968:
910:
252:
8487:
6464:
5292:
be the sums of the bundles of differential forms with coefficients in
2834:
172:
9022:"The Atiyah–Singer Index Theorem: What it is and why you should care"
8470:
7671:(1967), "A Lefschetz Fixed Point Formula for Elliptic Complexes: I",
6604:
embeds in, this reduces the index theorem to the case of spheres. If
3226:) do not admit local boundary conditions. To handle these operators,
2943:
2808:
2689:
192:
119:(related to the dimension of the space of solutions) is equal to the
9081:
8709:(1971), "Future extensions of index theory and elliptic operators",
8537:(1969), "On the triangulation of manifolds and the Hauptvermutung",
7713:
7684:
7618:
7580:
7542:
7504:
7438:
2693:
209:
published his results on the topological invariance of the rational
143:
The index problem for elliptic differential operators was posed by
8479:
8421:
8406:
8382:
Invariance Theory, the Heat Equation, and the Atiyah–Singer Theorem
6566:
5351:{\displaystyle {\overline {\partial }}+{\overline {\partial }}^{*}}
3218:
1030:
1017:, defined as the difference between the (finite) dimension of the
361:
196:
8976:(2nd ed.), Somerville, Mass.: International Press of Boston,
8866:
Teleman, N. (1984), "The index theorem on topological manifolds",
6553:
is a smooth vector bundle on the compact smooth oriented manifold
3689:
over the manifold. The index formula for this operator yields the
3007:
2605:
elliptic operators whose index does not vanish in odd dimensions.
276:
structures on topological manifolds of dimension different from 4.
8165:"Cyclic cohomology, the Novikov conjecture and hyperbolic groups"
7202:
6271:
7461:(1963), "The Index of Elliptic Operators on Compact Manifolds",
2949:
951:) is invertible for all non-zero cotangent vectors at any point
3289:, with terms coming from fixed-point submanifolds of the group
305:
Teleman establishes the index theorem on topological manifolds.
4208:{\textstyle e(TM)=\prod _{i}^{r}x_{i}(TM\otimes \mathbb {C} )}
3072:
Sullivan's result on Lipschitz and quasiconformal structures (
8302:
reprinted in volume 1 of his collected works, p. 65–75,
8072:, École Normale Supérieure, Secrétariat mathématique, Paris,
7178:
6876:
8896:
Teleman, N. (1985), "Transversality and the index theorem",
7082:
5525:
th cohomology group is just the coherent cohomology group H(
2540:
is an elliptic differential operator between vector bundles
1402:{\displaystyle \operatorname {ch} (D)\operatorname {Td} (X)}
770:, and so is elliptic as this is nonzero whenever any of the
8626:"Topological invariance of the rational Pontrjagin classes"
7190:
7022:
6962:
6904:
6902:
3254:. In this case the index is an element of the K-theory of
8828:"The index of signature operators on Lipschitz manifolds"
8674:, Lecture Notes in Mathematics, vol. 638, Springer,
7225:, Lecture Notes in Mathematics, vol. 638, Springer,
6262:
states that the signature of a compact oriented manifold
3960:{\displaystyle x_{i}(E\otimes \mathbb {C} )=c_{1}(l_{i})}
2253:-dimensional orientable (compact) manifold with non-zero
882:, as the symbol vanishes for some non-zero values of the
417:
is a differential operator on a Euclidean space of order
8020:
Berline, Nicole; Getzler, Ezra; Vergne, Michèle (1992),
7516:
This gives a proof using K-theory instead of cohomology.
7295:"algebraic topology - How to understand the Todd class?"
6974:
2608:
6938:
6899:
6881:
Pages displaying short descriptions of redirect targets
3346:
to derive this index theorem on spaces equipped with a
1079:
Suppose that the manifold is the circle (thought of as
179:(which was rediscovered by Atiyah and Singer in 1961).
7783:
Atiyah, M.; Bott, R.; Patodi, V. K. (1975), "Errata",
7740:(1973), "On the heat equation and the index theorem",
5669:
4141:
2909:
is a point, then we recover the statement above. Here
8199:
7871:
Atiyah, Michael; Schmid, Wilfried (1979), "Erratum",
7870:
7208:
6926:
6704:
6359:
6303:
6164:
5744:
5550:
5500:
5377:
5317:
4913:
4224:
4098:
3973:
3900:
3803:
3755:
3726:
3706:
3671:
3638:
3615:
3588:
3561:
3522:
3495:
3468:
3439:
3419:
3135:
3080:
3046:
3042:
3017:
2915:
2889:
2658:
2623:
2497:
then there is a pushforward (or "shriek") map from K(
2466:
2421:
2301:
2262:
2236:
2216:
2191:
2162:
2131:
2102:
2073:
2043:
2013:
1958:
1888:
1810:
1751:
1635:
1532:
1500:
1478:
1442:
1419:
1368:
1225:
1202:
1182:
1162:
1142:
1122:
862:
803:
776:
720:
678:
640:
605:
567:
550:{\displaystyle x_{1},\dots ,x_{k},y_{1},\dots ,y_{k}}
485:
431:
9054:
8019:
7968:
7106:
7070:
6652:
2559:
is the following: compute the (analytical) index of
2509:) is defined to be the image of this operation with
408:
7154:
7142:
7118:
7094:
3685:, and the topological index is the integral of the
9098:"Interview with Michael Atiyah and Isadore Singer"
9078:"Lecture notes on the Atiyah–Singer Index Theorem"
8721:(1979), "Hyperbolic geometry and homeomorphisms",
8131:
7166:
7130:
7058:
7046:
7010:
6839:
6647:) gave a new proof of the index theorem using the
6413:
6331:
6231:
6140:
5727:
5652:
5533:), so the analytical index of this complex is the
5513:
5483:
5350:
5304:even or odd, and we let the differential operator
5263:
5245:
4893:
4207:
4124:
4084:
3959:
3886:
3789:
3741:
3712:
3677:
3653:
3621:
3601:
3574:
3547:
3508:
3481:
3454:
3425:
3205:
2930:
2901:
2872:
2641:
2478:
2449:
2404:
2280:
2245:
2222:
2197:
2177:
2146:
2117:
2088:
2059:
2029:
1999:
1944:
1869:
1795:
1733:
1619:
1518:
1484:
1460:
1425:
1401:
1348:
1208:
1188:
1168:
1148:
1128:
874:
848:
789:
762:
691:
664:
618:
591:
549:
463:
8236:
7184:
6950:
6914:
4215:. As for the Chern character and the Todd class,
3035:
3013:
557:, given by dropping all terms of order less than
9120:
9045:: CS1 maint: bot: original URL status unknown (
8574:
7782:
7728:
7527:(1968b), "The Index of Elliptic Operators III",
7262:
7088:
7034:
6644:
6458:
3266:to the complex K-theory is not always injective.
2517:) can be naturally identified with the integers
889:The symbol of a differential operator of order
626:. So the symbol is homogeneous in the variables
259:gave a new proof of the index theorem using the
8746:
8558:
8529:
8159:
8064:
7565:(1971a), "The Index of Elliptic Operators IV",
7425:(1968), "The Index of Elliptic Operators: II",
7383:(1970), "Global Theory of Elliptic Operators",
7196:
7028:
6968:
6565:Atiyah and Singer's first published proof used
6483:
6253:
3008:Connes–Donaldson–Sullivan–Teleman index theorem
2998:
921:is a section of the pullback of the bundle Hom(
703:if the symbol is nonzero whenever at least one
151:. Some of the motivating examples included the
8968:
8725:, New York: Academic Press, pp. 543–595,
7603:(1971b), "The Index of Elliptic Operators V",
7491:(1968a), "The Index of Elliptic Operators I",
6451:, and its topological index is the L genus of
3116:The Atiyah–Singer theorem applies to elliptic
3107:for which the index theorem is known to hold.
3030:This theory is based on a signature operator
9027:. Archived from the original on June 24, 2006
7595:
7557:
7519:
7483:
7004:
7000:
6996:
6992:
6980:
6858:, which can be used to evaluate the limit as
3894:. Therefore, we can consider the Chern roots
3408:
3285:. Moreover, one gets generalizations of the
2950:Extensions of the Atiyah–Singer index theorem
2505:). The topological index of an element of K(
797:'s are nonzero. The wave operator has symbol
9055:Voitsekhovskii, M.I.; Shubin, M.A. (2001) ,
8448:Hilsum, M. (1999), "Structures riemaniennes
8273:
7820:
7453:
6944:
6908:
3433:is a compact oriented manifold of dimension
3316:
2586:that we can usually at least evaluate their
849:{\displaystyle -y_{1}^{2}+\cdots +y_{k}^{2}}
331:defined on differential forms of degree two.
131:, as special cases, and has applications to
27:Mathematical result in differential geometry
9095:
7417:
6573:is any inclusion of compact manifolds from
6526:The initial proof was based on that of the
2533:under this map "is" the topological index.
1945:{\displaystyle d(p^{*}E,p^{*}F,\sigma (D))}
763:{\displaystyle y_{1}^{2}+\cdots +y_{k}^{2}}
8939:(1982), "Supersymmetry and Morse theory",
8652:Seminar on the Atiyah–Singer Index Theorem
7697:
7663:
7633:
6151:Applying the index theorem, we obtain the
5728:{\textstyle e(TX)=\prod _{i}^{n}x_{i}(TX)}
388:is an elliptic differential operator from
9096:Raussen, Martin; Skau, Christian (2005),
8952:
8879:
8679:
8550:
8469:
8429:
8420:
8405:
8370:
8257:
8220:
8183:
8055:
7892:
7804:
7654:
7474:
7230:
6581:, they defined a 'pushforward' operation
5942:
5208:
5112:
4880:
4796:
4766:
4685:
4594:
4515:
4440:
4361:
4311:
4198:
4118:
4111:
4075:
4000:
3921:
3811:
3776:
3769:
3338:, at the suggestion of his Ph.D. advisor
2852:
2734:
1860:
1833:
1724:
1662:
966:on a compact manifold has a (non-unique)
634:. The symbol is well defined even though
8717:
8669:
8501:
8399:
7945:Collected works. Vol. 4. Index theory: 2
7915:Collected works. Vol. 3. Index theory: 1
7339:
7220:
7112:
7076:
6932:
6670:is a differential operator with adjoint
3516:to be the sum of the odd powers, define
3073:
2994:
2954:
1029:= 0), and the (finite) dimension of the
665:{\displaystyle \partial /\partial x_{i}}
592:{\displaystyle \partial /\partial x_{i}}
285:motivated by ideas of Edward Witten and
9134:Elliptic partial differential equations
8895:
8865:
8825:
8789:
8620:
8599:
8356:
8313:
8089:"Non-commutative differential geometry"
7939:
7908:
7371:
7367:
7160:
7148:
7124:
7100:
7016:
6656:
5514:{\displaystyle {\overline {\partial }}}
3790:{\displaystyle l_{1},\,\ldots ,\,l_{r}}
3240:
3061:
2990:
2986:
2982:
2964:
2960:
2493:is a compact submanifold of a manifold
293:; this covers many of the useful cases.
14:
9121:
9089:
9072:
9013:
8935:
8898:Integral Equations and Operator Theory
8705:
8645:
8603:The Atiyah–Patodi–Singer Index Theorem
8447:
8414:
8378:
8244:(1989), "Quasiconformal 4-manifolds",
8126:
8083:
8038:
7398:
7379:
7172:
7136:
7064:
7052:
6956:
6920:
6660:
3575:{\displaystyle \Lambda ^{\text{even}}}
3482:{\displaystyle \Lambda ^{\text{even}}}
3300:
3100:
3057:
2973:
2415:where division makes sense by pulling
1519:{\displaystyle \operatorname {ch} (D)}
1472:of the complexified tangent bundle of
1461:{\displaystyle \operatorname {Td} (X)}
1005:As the elliptic differential operator
933:. The differential operator is called
7982:"Riemann-Roch for singular varieties"
7480:An announcement of the index theorem.
3720:is a real vector bundle of dimension
3602:{\displaystyle \Lambda ^{\text{odd}}}
3509:{\displaystyle \Lambda ^{\text{odd}}}
2609:Relation to Grothendieck–Riemann–Roch
2457:back from the cohomology ring of the
1116:of an elliptic differential operator
8923:
8504:Math. USSR Izvestija (Engl. Transl.)
7329:Some Remarks on the Paper of Callias
7040:
6879: – Term in quantum field theory
6653:Berline, Getzler & Vergne (1992)
6596:that preserves the index. By taking
3065:
2972:For any abstract elliptic operator (
2601:has odd dimension, though there are
1107:
1095:) if λ is an integral multiple of 2π
7344:, Institute of Physics Publishing,
7209:Connes, Sullivan & Teleman 1994
6888: – Modified partition function
6655:. The proof is also published in (
6478:
6471:In dimension 4 this result implies
3110:
3081:Connes, Sullivan & Teleman 1994
3047:Connes, Sullivan & Teleman 1994
3043:Connes, Sullivan & Teleman 1994
3018:Connes, Sullivan & Teleman 1994
1000:
314:publishes his fundamental paper on
24:
9019:
6366:
5801:
5503:
5441:
5397:
5334:
5320:
4414:
4335:
4256:
4243:
3590:
3563:
3497:
3470:
3245:Atiyah–Patodi–Singer index theorem
2580:is equal to its topological index.
2513:some Euclidean space, for which K(
1433:up to a difference of sign. Here,
649:
641:
576:
568:
464:{\displaystyle x_{1},\dots ,x_{k}}
263:, described in a paper by Melrose.
171:had proved the integrality of the
25:
9150:
9139:Theorems in differential geometry
9008:
8430:Higson, Nigel; Roe, John (2000),
8277:(1960), "On elliptic equations",
6447:is the signature of the manifold
5280:with a holomorphic vector bundle
4125:{\displaystyle i=1,\,\ldots ,\,r}
2007:associated to two vector bundles
710:Example: The Laplace operator in
409:Symbol of a differential operator
76:Grothendieck–Riemann–Roch theorem
8022:Heat Kernels and Dirac Operators
6630:
6616:, then any elliptic operator on
6394:
6390:
6381:
5535:holomorphic Euler characteristic
3609:. Then the analytical index of
3099:+1)/2, introduced by M. Hilsum (
1796:{\displaystyle p:B(X)/S(X)\to X}
8672:The Atiyah-Singer Index Theorem
8552:10.1090/S0002-9904-1969-12271-8
8524:10.1070/IM1975v009n04ABEH001497
8299:10.1070/rm1960v015n03ABEH004094
7656:10.1090/S0002-9904-1966-11483-0
7476:10.1090/S0002-9904-1963-10957-X
7333:
7322:
7311:
7287:
7256:
7223:The Atiyah-Singer Index Theorem
7214:
6528:Hirzebruch–Riemann–Roch theorem
6153:Hirzebruch-Riemann-Roch theorem
5494:with the differential given by
5264:Hirzebruch–Riemann–Roch theorem
3321:discrete series representations
3079:The quasiconformal structures (
1952:is the "difference element" in
1013:. Any Fredholm operator has an
195:theory of the first proof with
157:Hirzebruch–Riemann–Roch theorem
9111:R. R. Seeley and other (1999)
8586:, Princeton University Press,
7342:Geometry, topology and physics
7274:, Princeton University Press,
7089:Atiyah, Bott & Patodi 1973
6748:
6735:
6717:
6711:
6324:
6312:
6226:
6217:
6208:
6202:
6180:
6168:
6131:
6122:
5967:
5958:
5946:
5929:
5916:
5907:
5856:
5850:
5785:
5775:
5722:
5713:
5682:
5673:
5647:
5635:
5626:
5614:
5589:
5579:
5563:
5557:
5475:
5472:
5466:
5431:
5428:
5422:
5387:
5381:
5240:
5231:
5212:
5195:
5161:
5151:
5132:
5122:
5116:
5099:
4939:
4929:
4923:
4917:
4884:
4867:
4800:
4783:
4770:
4753:
4689:
4672:
4620:
4610:
4598:
4581:
4519:
4502:
4391:
4381:
4315:
4291:
4202:
4185:
4154:
4145:
4079:
4065:
4004:
3990:
3954:
3941:
3925:
3911:
3648:
3642:
3319:to rederive properties of the
3197:
3184:
3178:
3165:
3152:
3139:
2925:
2919:
2856:
2842:
2821:
2815:
2798:
2792:
2738:
2724:
2705:
2699:
2679:
2673:
2633:
2435:
2425:
2399:
2393:
2381:
2372:
2364:
2358:
2346:
2340:
2312:
2302:
2275:
2266:
2172:
2166:
2141:
2135:
2112:
2106:
2083:
2077:
1994:
1991:
1985:
1974:
1968:
1962:
1939:
1936:
1930:
1892:
1864:
1850:
1837:
1826:
1820:
1787:
1784:
1778:
1767:
1761:
1728:
1717:
1711:
1700:
1694:
1688:
1669:
1666:
1652:
1614:
1611:
1608:
1605:
1599:
1561:
1555:
1546:
1513:
1507:
1455:
1449:
1396:
1390:
1381:
1375:
1343:
1337:
1328:
1322:
1297:
1287:
1281:
1275:
1272:
1266:
1257:
1251:
1236:
1226:
1196:-dimensional compact manifold
1136:between smooth vector bundles
109:elliptic differential operator
13:
1:
8138:, San Diego: Academic Press,
7361:
7318:Index Theorems on Open Spaces
7185:Donaldson & Sullivan 1989
6459:Â genus and Rochlin's theorem
4904:Applying the index theorem,
3287:Lefschetz fixed-point theorem
3036:Donaldson & Sullivan 1989
3014:Donaldson & Sullivan 1989
2997:) and topological cobordism (
2125:between them on the subspace
1065:This is sometimes called the
1009:has a pseudoinverse, it is a
8974:The Founders of Index Theory
8606:, Wellesley, Mass.: Peters,
8600:Melrose, Richard B. (1993),
8372:10.1016/0040-9383(86)90008-X
8222:10.1016/0040-9383(94)90003-5
8185:10.1016/0040-9383(90)90003-3
8057:10.1016/0022-1236(84)90101-0
6892:
6536:pseudodifferential operators
6521:
6484:Pseudodifferential operators
6260:Hirzebruch signature theorem
6254:Hirzebruch signature theorem
5990:
5827:
5506:
5337:
5323:
5284:. We let the vector bundles
4038:
3879:
3843:
3374:is even, it is always zero.
3370:−1)-sphere at infinity. If
3118:pseudodifferential operators
2000:{\displaystyle K(B(X)/S(X))}
929:) to the cotangent space of
161:Hirzebruch signature theorem
80:Hirzebruch signature theorem
7:
9062:Encyclopedia of Mathematics
8929:Symp. Int. Top. Alg. Mexico
8434:, Oxford University Press,
7197:Connes & Moscovici 1990
7029:Kirby & Siebenmann 1969
7005:Atiyah & Singer (1971b)
7001:Atiyah & Singer (1971a)
6997:Atiyah & Singer (1968b)
6993:Atiyah & Singer (1968a)
6870:
6560:
6490:pseudodifferential operator
3555:, considered as a map from
3403:
2999:Kirby & Siebenmann 1977
1053:) = dim Ker(D) − dim Coker(
995:pseudodifferential operator
856:, which is not elliptic if
347:
155:and its generalization the
107:(1963), states that for an
97:Atiyah–Singer index theorem
31:Atiyah–Singer index theorem
10:
9155:
8634:Doklady Akademii Nauk SSSR
7299:Mathematics Stack Exchange
6691:. Therefore, the index of
6612:is some point embedded in
6569:rather than cobordism. If
6487:
6332:{\displaystyle i^{k(k-1)}}
3691:Chern–Gauss–Bonnet theorem
3409:Chern-Gauss-Bonnet theorem
3317:Atiyah & Schmid (1977)
2993:), Kasparov's K-homology (
2450:{\displaystyle e(TX)^{-1}}
2178:{\displaystyle \sigma (D)}
2118:{\displaystyle \sigma (D)}
1411:fundamental homology class
223:'s results, combined with
138:
125:Chern–Gauss–Bonnet theorem
72:Chern–Gauss–Bonnet theorem
8379:Gilkey, Peter B. (1994),
6981:Atiyah & Singer 1968a
6592:to elliptic operators of
6588:on elliptic operators of
3548:{\displaystyle D=d+d^{*}}
3295:equivariant index theorem
3083:) and more generally the
2615:Grothendieck–Riemann–Roch
1057:) = dim Ker(D) − dim Ker(
67:
59:
45:
35:
8792:Inventiones Mathematicae
8711:Prospects in Mathematics
8580:Michelsohn, Marie-Louise
8163:; Moscovici, H. (1990),
7340:Nakahara, Mikio (2003),
7268:Michelsohn, Marie-Louise
6909:Atiyah & Singer 1963
3654:{\displaystyle \chi (M)}
3277:on the compact manifold
3054:Prospects in Mathematics
2642:{\displaystyle f:X\to Y}
2576:The analytical index of
2548:over a compact manifold
905:, homogeneous of degree
8987:- Personal accounts on
8134:Noncommutative Geometry
7387:, University of Tokio,
6600:to be some sphere that
5276:of (complex) dimension
5258:Chern-Weil homomorphism
875:{\displaystyle k\geq 2}
316:noncommutative geometry
9129:Differential operators
8954:10.4310/jdg/1214437492
8750:; Teleman, N. (1983),
8539:Bull. Amer. Math. Soc.
7463:Bull. Amer. Math. Soc.
6841:
6455:, so these are equal.
6443:The analytic index of
6415:
6333:
6233:
6142:
6077:
6020:
5873:
5774:
5729:
5702:
5654:
5515:
5485:
5352:
5247:
5184:
5056:
5028:
4975:
4895:
4824:
4716:
4481:
4209:
4174:
4126:
4086:
3961:
3888:
3791:
3743:
3714:
3679:
3655:
3623:
3603:
3576:
3549:
3510:
3483:
3456:
3427:
3207:
2932:
2903:
2874:
2835:
2717:
2643:
2563:using only the symbol
2480:
2451:
2406:
2282:
2247:
2224:
2199:
2179:
2148:
2119:
2090:
2061:
2060:{\displaystyle p^{*}F}
2031:
2030:{\displaystyle p^{*}E}
2001:
1946:
1871:
1797:
1745:for the sphere bundle
1735:
1621:
1520:
1486:
1462:
1427:
1403:
1350:
1210:
1190:
1170:
1150:
1130:
937:if the element of Hom(
876:
850:
791:
764:
693:
672:does not commute with
666:
620:
593:
551:
465:
400:to smooth sections of
149:topological invariants
8670:Shanahan, P. (1978),
8617:Free online textbook.
8458:Annals of Mathematics
7701:Annals of Mathematics
7673:Annals of Mathematics
7606:Annals of Mathematics
7568:Annals of Mathematics
7530:Annals of Mathematics
7493:Annals of Mathematics
7427:Annals of Mathematics
7221:Shanahan, P. (1978),
6842:
6530:(1954), and involved
6416:
6334:
6234:
6143:
6063:
6006:
5859:
5760:
5730:
5688:
5655:
5516:
5486:
5353:
5248:
5170:
5042:
5014:
4961:
4896:
4810:
4702:
4467:
4210:
4160:
4127:
4087:
3962:
3889:
3792:
3744:
3715:
3680:
3656:
3624:
3604:
3577:
3550:
3511:
3484:
3457:
3428:
3332:Callias index theorem
3325:semisimple Lie groups
3208:
2955:Teleman index theorem
2933:
2904:
2875:
2804:
2685:
2644:
2481:
2452:
2407:
2283:
2281:{\displaystyle e(TX)}
2248:
2225:
2200:
2180:
2149:
2120:
2091:
2062:
2032:
2002:
1947:
1872:
1798:
1736:
1622:
1521:
1487:
1463:
1428:
1404:
1351:
1211:
1191:
1171:
1151:
1131:
893:on a smooth manifold
877:
851:
792:
790:{\displaystyle y_{i}}
765:
714:variables has symbol
694:
692:{\displaystyle x_{i}}
667:
621:
619:{\displaystyle y_{i}}
594:
552:
466:
221:Laurent C. Siebenmann
93:differential geometry
40:Differential geometry
8826:Teleman, N. (1983),
8024:, Berlin: Springer,
7643:Bull. Am. Math. Soc.
6969:Cartan-Schwartz 1965
6702:
6357:
6301:
6162:
5742:
5667:
5548:
5498:
5375:
5315:
4911:
4222:
4139:
4096:
3971:
3898:
3801:
3753:
3742:{\displaystyle n=2r}
3724:
3704:
3669:
3636:
3631:Euler characteristic
3613:
3586:
3559:
3520:
3493:
3466:
3455:{\displaystyle n=2r}
3437:
3417:
3395:Robert Thomas Seeley
3385:, as generalized by
3283:equivariant K-theory
3133:
3068:) and index theory.
2931:{\displaystyle K(X)}
2913:
2887:
2656:
2621:
2529:), and the image in
2464:
2419:
2299:
2288:, then applying the
2260:
2234:
2214:
2189:
2160:
2147:{\displaystyle S(X)}
2129:
2100:
2089:{\displaystyle B(X)}
2071:
2041:
2011:
1956:
1886:
1808:
1749:
1633:
1530:
1498:
1476:
1440:
1417:
1366:
1223:
1200:
1180:
1160:
1140:
1120:
860:
801:
774:
718:
676:
638:
603:
565:
483:
475:is the function of 2
429:
213:on smooth manifolds.
189:Princeton University
165:Friedrich Hirzebruch
153:Riemann–Roch theorem
129:Riemann–Roch theorem
9090:Links of interviews
9014:Links on the theory
8804:1980InMat..61..227T
8516:1975IzMat...9..751K
8432:Analytic K-homology
8335:1983CMaPh..92..163G
8323:Commun. Math. Phys.
8291:1960RuMaS..15..113G
8040:Bismut, Jean-Michel
7885:1979InMat..54..189A
7842:1977InMat..42....1A
7797:1975InMat..28..277A
7754:1973InMat..19..279A
6620:is the image under
6434:exterior derivative
6341:Hodge star operator
3698:splitting principle
3389:, was published by
3336:Constantine Callias
3306:von Neumann algebra
2902:{\displaystyle Y=*}
2833:
2716:
2692:
2479:{\displaystyle BSO}
2096:and an isomorphism
845:
821:
759:
735:
364:(without boundary).
133:theoretical physics
32:
9107:, pp. 223–231
9084:on March 29, 2017.
9074:Wassermann, Antony
8910:10.1007/BF01201710
8881:10.1007/BF02392376
8844:10.1007/BF02953772
8812:10.1007/BF01390066
8768:10.1007/BF02953773
8762:, Paris: 291–293,
8690:10.1007/BFb0068264
8647:Palais, Richard S.
8343:10.1007/BF01210843
8259:10.1007/BF02392736
8105:10.1007/BF02698807
7998:10.1007/BF02684299
7894:10.1007/BF01408936
7850:10.1007/BF01389783
7806:10.1007/BF01425562
7762:10.1007/BF01425417
7601:Singer, Isadore M.
7597:Atiyah, Michael F.
7563:Singer, Isadore M.
7559:Atiyah, Michael F.
7525:Singer, Isadore M.
7521:Atiyah, Michael F.
7489:Singer, Isadore M.
7485:Atiyah, Michael F.
7459:Singer, Isadore M.
7455:Atiyah, Michael F.
7241:10.1007/BFb0068264
6837:
6496:Fourier transforms
6440:* is its adjoint.
6411:
6329:
6276:signature operator
6246:complex manifolds
6229:
6138:
6136:
5725:
5650:
5578:
5511:
5481:
5348:
5243:
4891:
4889:
4540:
4205:
4122:
4082:
3957:
3884:
3787:
3739:
3710:
3675:
3651:
3619:
3599:
3572:
3545:
3506:
3479:
3452:
3423:
3315:, and was used by
3224:signature operator
3203:
2940:Grothendieck group
2928:
2899:
2870:
2868:
2639:
2603:pseudodifferential
2476:
2447:
2402:
2278:
2246:{\displaystyle 2m}
2243:
2220:
2195:
2175:
2144:
2115:
2086:
2057:
2027:
1997:
1942:
1867:
1793:
1731:
1617:
1516:
1482:
1458:
1423:
1399:
1346:
1206:
1186:
1166:
1146:
1126:
872:
846:
831:
807:
787:
760:
745:
721:
689:
662:
616:
589:
547:
461:
325:Simon K. Donaldson
287:Luis Alvarez-Gaume
211:Pontryagin classes
30:
9051:Pdf presentation.
8732:978-0-12-158860-1
8699:978-0-387-08660-6
8662:978-0-691-08031-4
8613:978-1-56881-002-7
8392:978-0-8493-7874-4
8279:Russ. Math. Surv.
8145:978-0-12-185860-5
8031:978-3-540-53340-5
7954:978-0-19-853278-1
7925:978-0-19-853277-4
7704:, Second Series,
7675:, Second series,
7609:, Second Series,
7571:, Second Series,
7533:, Second Series,
7429:, Second Series,
7250:978-0-387-08660-6
6850:for any positive
6473:Rochlin's theorem
6120:
6061:
5993:
5830:
5569:
5509:
5340:
5326:
5097:
5040:
4865:
4525:
4262:
4249:
4041:
3882:
3846:
3713:{\displaystyle E}
3678:{\displaystyle M}
3622:{\displaystyle D}
3596:
3569:
3503:
3476:
3426:{\displaystyle M}
2830:
2813:
2714:
2697:
2459:classifying space
2385:
2223:{\displaystyle X}
2198:{\displaystyle D}
2185:is the symbol of
1485:{\displaystyle X}
1426:{\displaystyle X}
1209:{\displaystyle X}
1189:{\displaystyle n}
1169:{\displaystyle F}
1149:{\displaystyle E}
1129:{\displaystyle D}
1114:topological index
1108:Topological index
1011:Fredholm operator
207:Sergey P. Novikov
121:topological index
89:
88:
84:Rokhlin's theorem
16:(Redirected from
9146:
9108:
9102:
9085:
9080:. Archived from
9069:
9057:"Index formulas"
9050:
9044:
9036:
9034:
9032:
9026:
8986:
8965:
8956:
8932:
8931:, pp. 54–67
8920:
8892:
8883:
8868:Acta Mathematica
8862:
8822:
8786:
8743:
8714:
8702:
8683:
8665:
8642:
8630:
8616:
8596:
8576:Lawson, H. Blane
8571:
8564:Siebenmann, L.C.
8555:
8554:
8535:Siebenmann, L.C.
8526:
8498:
8473:
8464:(3): 1007–1022,
8444:
8426:
8424:
8411:
8409:
8395:
8375:
8374:
8353:
8301:
8270:
8261:
8246:Acta Mathematica
8233:
8224:
8196:
8187:
8169:
8156:
8137:
8123:
8080:
8060:
8059:
8034:
8016:
7986:Acta Mathematica
7965:
7936:
7905:
7896:
7868:
7826:Schmid, Wilfried
7817:
7808:
7780:
7724:
7695:
7659:
7658:
7629:
7591:
7553:
7515:
7479:
7478:
7449:
7414:
7395:
7355:
7354:
7337:
7331:
7326:
7320:
7315:
7309:
7308:
7306:
7305:
7291:
7285:
7284:
7264:Lawson, H. Blane
7260:
7254:
7253:
7234:
7218:
7212:
7206:
7200:
7194:
7188:
7182:
7176:
7170:
7164:
7158:
7152:
7146:
7140:
7134:
7128:
7122:
7116:
7110:
7104:
7098:
7092:
7086:
7080:
7074:
7068:
7062:
7056:
7050:
7044:
7038:
7032:
7026:
7020:
7014:
7008:
6990:
6984:
6978:
6972:
6966:
6960:
6954:
6948:
6942:
6936:
6930:
6924:
6918:
6912:
6906:
6882:
6846:
6844:
6843:
6838:
6836:
6832:
6831:
6830:
6829:
6793:
6789:
6788:
6784:
6783:
6747:
6746:
6608:is a sphere and
6532:cobordism theory
6479:Proof techniques
6420:
6418:
6417:
6412:
6410:
6409:
6404:
6400:
6399:
6398:
6397:
6384:
6373:
6338:
6336:
6335:
6330:
6328:
6327:
6270:is given by the
6238:
6236:
6235:
6230:
6195:
6194:
6147:
6145:
6144:
6139:
6137:
6121:
6119:
6118:
6117:
6116:
6115:
6094:
6093:
6092:
6079:
6076:
6071:
6062:
6060:
6059:
6058:
6057:
6056:
6032:
6031:
6022:
6019:
6014:
5998:
5994:
5989:
5981:
5945:
5906:
5902:
5901:
5900:
5899:
5898:
5872:
5867:
5836:
5832:
5831:
5826:
5822:
5821:
5811:
5809:
5808:
5793:
5792:
5773:
5768:
5734:
5732:
5731:
5726:
5712:
5711:
5701:
5696:
5659:
5657:
5656:
5651:
5613:
5612:
5597:
5596:
5577:
5520:
5518:
5517:
5512:
5510:
5502:
5490:
5488:
5487:
5482:
5465:
5464:
5455:
5454:
5421:
5420:
5411:
5410:
5357:
5355:
5354:
5349:
5347:
5346:
5341:
5333:
5327:
5319:
5274:complex manifold
5252:
5250:
5249:
5244:
5227:
5226:
5211:
5194:
5193:
5183:
5178:
5169:
5168:
5150:
5149:
5140:
5139:
5115:
5098:
5096:
5095:
5094:
5093:
5092:
5068:
5067:
5058:
5055:
5050:
5041:
5039:
5038:
5037:
5027:
5022:
5012:
5011:
5007:
5006:
5005:
5004:
5003:
4974:
4969:
4959:
4957:
4956:
4947:
4946:
4900:
4898:
4897:
4892:
4890:
4883:
4866:
4864:
4863:
4862:
4861:
4860:
4836:
4835:
4826:
4823:
4818:
4799:
4769:
4752:
4748:
4747:
4746:
4745:
4744:
4715:
4710:
4695:
4688:
4671:
4670:
4669:
4668:
4648:
4647:
4646:
4645:
4628:
4627:
4597:
4580:
4579:
4578:
4577:
4560:
4559:
4558:
4557:
4539:
4518:
4501:
4500:
4499:
4498:
4480:
4475:
4454:
4450:
4449:
4448:
4444:
4443:
4432:
4431:
4422:
4421:
4399:
4398:
4371:
4370:
4369:
4365:
4364:
4353:
4352:
4343:
4342:
4314:
4303:
4302:
4271:
4270:
4269:
4265:
4264:
4263:
4260:
4251:
4250:
4247:
4214:
4212:
4211:
4206:
4201:
4184:
4183:
4173:
4168:
4131:
4129:
4128:
4123:
4091:
4089:
4088:
4083:
4078:
4064:
4063:
4048:
4047:
4046:
4042:
4037:
4036:
4027:
4019:
4018:
4003:
3989:
3988:
3966:
3964:
3963:
3958:
3953:
3952:
3940:
3939:
3924:
3910:
3909:
3893:
3891:
3890:
3885:
3883:
3878:
3877:
3868:
3863:
3862:
3847:
3842:
3841:
3832:
3827:
3826:
3814:
3796:
3794:
3793:
3788:
3786:
3785:
3765:
3764:
3748:
3746:
3745:
3740:
3719:
3717:
3716:
3711:
3684:
3682:
3681:
3676:
3663:Hodge cohomology
3660:
3658:
3657:
3652:
3628:
3626:
3625:
3620:
3608:
3606:
3605:
3600:
3598:
3597:
3594:
3581:
3579:
3578:
3573:
3571:
3570:
3567:
3554:
3552:
3551:
3546:
3544:
3543:
3515:
3513:
3512:
3507:
3505:
3504:
3501:
3488:
3486:
3485:
3480:
3478:
3477:
3474:
3461:
3459:
3458:
3453:
3432:
3430:
3429:
3424:
3348:Hermitian matrix
3212:
3210:
3209:
3204:
3196:
3195:
3177:
3176:
3164:
3163:
3151:
3150:
3127:elliptic complex
3111:Other extensions
2937:
2935:
2934:
2929:
2908:
2906:
2905:
2900:
2879:
2877:
2876:
2871:
2869:
2866:
2865:
2864:
2863:
2860:
2855:
2836:
2832:
2831:
2828:
2814:
2811:
2787:
2783:
2782:
2773:
2772:
2765:
2763:
2762:
2756:
2755:
2745:
2742:
2737:
2718:
2715:
2712:
2698:
2695:
2691:
2668:
2665:
2664:
2663:
2662:
2648:
2646:
2645:
2640:
2485:
2483:
2482:
2477:
2456:
2454:
2453:
2448:
2446:
2445:
2411:
2409:
2408:
2403:
2386:
2384:
2367:
2332:
2330:
2329:
2320:
2319:
2290:Thom isomorphism
2287:
2285:
2284:
2279:
2252:
2250:
2249:
2244:
2229:
2227:
2226:
2221:
2204:
2202:
2201:
2196:
2184:
2182:
2181:
2176:
2153:
2151:
2150:
2145:
2124:
2122:
2121:
2116:
2095:
2093:
2092:
2087:
2066:
2064:
2063:
2058:
2053:
2052:
2036:
2034:
2033:
2028:
2023:
2022:
2006:
2004:
2003:
1998:
1981:
1951:
1949:
1948:
1943:
1920:
1919:
1904:
1903:
1876:
1874:
1873:
1868:
1863:
1849:
1848:
1836:
1802:
1800:
1799:
1794:
1774:
1743:Thom isomorphism
1740:
1738:
1737:
1732:
1727:
1707:
1687:
1686:
1665:
1651:
1650:
1626:
1624:
1623:
1618:
1589:
1588:
1573:
1572:
1545:
1544:
1525:
1523:
1522:
1517:
1491:
1489:
1488:
1483:
1467:
1465:
1464:
1459:
1432:
1430:
1429:
1424:
1413:of the manifold
1408:
1406:
1405:
1400:
1361:cohomology class
1355:
1353:
1352:
1347:
1315:
1314:
1305:
1304:
1244:
1243:
1215:
1213:
1212:
1207:
1195:
1193:
1192:
1187:
1175:
1173:
1172:
1167:
1155:
1153:
1152:
1147:
1135:
1133:
1132:
1127:
1067:analytical index
1001:Analytical index
899:cotangent bundle
881:
879:
878:
873:
855:
853:
852:
847:
844:
839:
820:
815:
796:
794:
793:
788:
786:
785:
769:
767:
766:
761:
758:
753:
734:
729:
698:
696:
695:
690:
688:
687:
671:
669:
668:
663:
661:
660:
648:
625:
623:
622:
617:
615:
614:
598:
596:
595:
590:
588:
587:
575:
556:
554:
553:
548:
546:
545:
527:
526:
514:
513:
495:
494:
470:
468:
467:
462:
460:
459:
441:
440:
117:analytical index
113:compact manifold
33:
29:
21:
9154:
9153:
9149:
9148:
9147:
9145:
9144:
9143:
9119:
9118:
9100:
9092:
9038:
9037:
9030:
9028:
9024:
9016:
9011:
9006:
8984:
8972:, ed. (2009) ,
8733:
8700:
8681:10.1.1.193.9222
8663:
8628:
8614:
8594:
8442:
8393:
8275:Gel'fand, I. M.
8238:Donaldson, S.K.
8167:
8146:
8066:Cartan-Schwartz
8044:J. Funct. Anal.
8032:
7955:
7941:Atiyah, Michael
7926:
7910:Atiyah, Michael
7822:Atiyah, Michael
7714:10.2307/1970721
7685:10.2307/1970694
7619:10.2307/1970757
7581:10.2307/1970756
7543:10.2307/1970717
7505:10.2307/1970715
7439:10.2307/1970716
7364:
7359:
7358:
7352:
7338:
7334:
7327:
7323:
7316:
7312:
7303:
7301:
7293:
7292:
7288:
7282:
7261:
7257:
7251:
7232:10.1.1.193.9222
7219:
7215:
7207:
7203:
7195:
7191:
7183:
7179:
7171:
7167:
7159:
7155:
7147:
7143:
7135:
7131:
7123:
7119:
7111:
7107:
7099:
7095:
7087:
7083:
7075:
7071:
7063:
7059:
7051:
7047:
7039:
7035:
7027:
7023:
7015:
7011:
6991:
6987:
6979:
6975:
6967:
6963:
6955:
6951:
6943:
6939:
6931:
6927:
6919:
6915:
6907:
6900:
6895:
6880:
6873:
6825:
6821:
6811:
6807:
6803:
6779:
6775:
6768:
6764:
6760:
6742:
6738:
6703:
6700:
6699:
6633:
6626:
6587:
6563:
6524:
6492:
6486:
6481:
6461:
6405:
6393:
6389:
6388:
6380:
6379:
6375:
6374:
6369:
6358:
6355:
6354:
6349:Hodge Laplacian
6343:. The operator
6308:
6304:
6302:
6299:
6298:
6293:, that acts on
6256:
6190:
6186:
6163:
6160:
6159:
6135:
6134:
6111:
6107:
6106:
6102:
6095:
6088:
6084:
6080:
6078:
6072:
6067:
6052:
6048:
6044:
6040:
6033:
6027:
6023:
6021:
6015:
6010:
5999:
5982:
5980:
5976:
5941:
5920:
5919:
5894:
5890:
5889:
5885:
5878:
5874:
5868:
5863:
5837:
5817:
5813:
5812:
5810:
5804:
5800:
5788:
5784:
5769:
5764:
5759:
5755:
5745:
5743:
5740:
5739:
5707:
5703:
5697:
5692:
5668:
5665:
5664:
5608:
5604:
5592:
5588:
5573:
5549:
5546:
5545:
5501:
5499:
5496:
5495:
5460:
5456:
5444:
5440:
5416:
5412:
5400:
5396:
5376:
5373:
5372:
5342:
5332:
5331:
5318:
5316:
5313:
5312:
5266:
5222:
5218:
5207:
5189:
5185:
5179:
5174:
5164:
5160:
5145:
5141:
5135:
5131:
5111:
5088:
5084:
5080:
5076:
5069:
5063:
5059:
5057:
5051:
5046:
5033:
5029:
5023:
5018:
5013:
4999:
4995:
4991:
4987:
4980:
4976:
4970:
4965:
4960:
4958:
4952:
4948:
4942:
4938:
4912:
4909:
4908:
4888:
4887:
4879:
4856:
4852:
4848:
4844:
4837:
4831:
4827:
4825:
4819:
4814:
4803:
4795:
4774:
4773:
4765:
4740:
4736:
4732:
4728:
4721:
4717:
4711:
4706:
4693:
4692:
4684:
4664:
4660:
4656:
4652:
4641:
4637:
4633:
4629:
4623:
4619:
4593:
4573:
4569:
4565:
4561:
4553:
4549:
4545:
4541:
4529:
4514:
4494:
4490:
4486:
4482:
4476:
4471:
4452:
4451:
4439:
4427:
4423:
4417:
4413:
4412:
4408:
4407:
4406:
4394:
4390:
4360:
4348:
4344:
4338:
4334:
4333:
4329:
4328:
4327:
4310:
4298:
4294:
4272:
4259:
4255:
4246:
4242:
4241:
4237:
4236:
4235:
4225:
4223:
4220:
4219:
4197:
4179:
4175:
4169:
4164:
4140:
4137:
4136:
4097:
4094:
4093:
4074:
4059:
4055:
4032:
4028:
4026:
4022:
4021:
4020:
4014:
4010:
3999:
3978:
3974:
3972:
3969:
3968:
3948:
3944:
3935:
3931:
3920:
3905:
3901:
3899:
3896:
3895:
3873:
3869:
3867:
3858:
3854:
3837:
3833:
3831:
3822:
3818:
3810:
3802:
3799:
3798:
3781:
3777:
3760:
3756:
3754:
3751:
3750:
3725:
3722:
3721:
3705:
3702:
3701:
3670:
3667:
3666:
3637:
3634:
3633:
3614:
3611:
3610:
3593:
3589:
3587:
3584:
3583:
3566:
3562:
3560:
3557:
3556:
3539:
3535:
3521:
3518:
3517:
3500:
3496:
3494:
3491:
3490:
3473:
3469:
3467:
3464:
3463:
3438:
3435:
3434:
3418:
3415:
3414:
3411:
3406:
3379:Hörmander index
3191:
3187:
3172:
3168:
3159:
3155:
3146:
3142:
3134:
3131:
3130:
3113:
3010:
2957:
2952:
2914:
2911:
2910:
2888:
2885:
2884:
2867:
2861:
2859:
2851:
2837:
2827:
2810:
2809:
2803:
2801:
2785:
2784:
2778:
2774:
2768:
2767:
2764:
2758:
2757:
2751:
2747:
2743:
2741:
2733:
2719:
2711:
2694:
2690:
2684:
2682:
2666:
2659:
2657:
2654:
2653:
2622:
2619:
2618:
2611:
2465:
2462:
2461:
2438:
2434:
2420:
2417:
2416:
2368:
2333:
2331:
2325:
2321:
2315:
2311:
2300:
2297:
2296:
2261:
2258:
2257:
2235:
2232:
2231:
2215:
2212:
2211:
2190:
2187:
2186:
2161:
2158:
2157:
2130:
2127:
2126:
2101:
2098:
2097:
2072:
2069:
2068:
2048:
2044:
2042:
2039:
2038:
2018:
2014:
2012:
2009:
2008:
1977:
1957:
1954:
1953:
1915:
1911:
1899:
1895:
1887:
1884:
1883:
1879:Chern character
1859:
1844:
1840:
1832:
1809:
1806:
1805:
1770:
1750:
1747:
1746:
1723:
1703:
1676:
1672:
1661:
1646:
1642:
1634:
1631:
1630:
1584:
1580:
1568:
1564:
1537:
1533:
1531:
1528:
1527:
1499:
1496:
1495:
1477:
1474:
1473:
1441:
1438:
1437:
1418:
1415:
1414:
1367:
1364:
1363:
1310:
1306:
1300:
1296:
1239:
1235:
1224:
1221:
1220:
1201:
1198:
1197:
1181:
1178:
1177:
1161:
1158:
1157:
1141:
1138:
1137:
1121:
1118:
1117:
1110:
1003:
949:
942:
861:
858:
857:
840:
835:
816:
811:
802:
799:
798:
781:
777:
775:
772:
771:
754:
749:
730:
725:
719:
716:
715:
683:
679:
677:
674:
673:
656:
652:
644:
639:
636:
635:
610:
606:
604:
601:
600:
583:
579:
571:
566:
563:
562:
541:
537:
522:
518:
509:
505:
490:
486:
484:
481:
480:
455:
451:
436:
432:
430:
427:
426:
411:
350:
291:Dirac operators
270:Dennis Sullivan
145:Israel Gel'fand
141:
82:
78:
74:
28:
23:
22:
15:
12:
11:
5:
9152:
9142:
9141:
9136:
9131:
9117:
9116:
9109:
9105:Notices of AMS
9091:
9088:
9087:
9086:
9070:
9052:
9020:Mazzeo, Rafe.
9015:
9012:
9010:
9009:External links
9007:
9005:
9004:
8983:978-1571461377
8982:
8970:Shing-Tung Yau
8966:
8947:(4): 661–692,
8941:J. Diff. Geom.
8937:Witten, Edward
8933:
8921:
8904:(5): 693–719,
8893:
8863:
8823:
8798:(3): 227–249,
8787:
8744:
8731:
8715:
8703:
8698:
8667:
8661:
8643:
8618:
8612:
8597:
8592:
8572:
8556:
8545:(4): 742–749,
8527:
8510:(4): 751–792,
8499:
8480:10.2307/121079
8445:
8440:
8427:
8412:
8397:
8391:
8376:
8354:
8329:(2): 163–178,
8311:
8285:(3): 113–123,
8271:
8234:
8215:(4): 663–681,
8197:
8178:(3): 345–388,
8157:
8144:
8124:
8081:
8062:
8036:
8030:
8017:
7978:Macpherson, R.
7966:
7953:
7937:
7924:
7906:
7879:(2): 189–192,
7818:
7791:(3): 277–280,
7748:(4): 279–330,
7726:
7708:(3): 451–491,
7679:(2): 374–407,
7661:
7631:
7613:(1): 139–149,
7593:
7575:(1): 119–138,
7555:
7537:(3): 546–604,
7517:
7499:(3): 484–530,
7481:
7469:(3): 422–433,
7451:
7433:(3): 531–545,
7415:
7396:
7376:
7363:
7360:
7357:
7356:
7350:
7332:
7321:
7310:
7286:
7280:
7255:
7249:
7213:
7201:
7189:
7177:
7165:
7153:
7141:
7129:
7117:
7105:
7093:
7081:
7069:
7057:
7045:
7033:
7021:
7009:
6985:
6973:
6961:
6949:
6937:
6925:
6913:
6897:
6896:
6894:
6891:
6890:
6889:
6883:
6872:
6869:
6848:
6847:
6835:
6828:
6824:
6820:
6817:
6814:
6810:
6806:
6802:
6799:
6796:
6792:
6787:
6782:
6778:
6774:
6771:
6767:
6763:
6759:
6756:
6753:
6750:
6745:
6741:
6737:
6734:
6731:
6728:
6725:
6722:
6719:
6716:
6713:
6710:
6707:
6639:, and
6632:
6629:
6624:
6585:
6562:
6559:
6523:
6520:
6488:Main article:
6485:
6482:
6480:
6477:
6460:
6457:
6432:is the Cartan
6424:restricted to
6422:
6421:
6408:
6403:
6396:
6392:
6387:
6383:
6378:
6372:
6368:
6365:
6362:
6326:
6323:
6320:
6317:
6314:
6311:
6307:
6266:of dimension 4
6255:
6252:
6240:
6239:
6228:
6225:
6222:
6219:
6216:
6213:
6210:
6207:
6204:
6201:
6198:
6193:
6189:
6185:
6182:
6179:
6176:
6173:
6170:
6167:
6149:
6148:
6133:
6130:
6127:
6124:
6114:
6110:
6105:
6101:
6098:
6091:
6087:
6083:
6075:
6070:
6066:
6055:
6051:
6047:
6043:
6039:
6036:
6030:
6026:
6018:
6013:
6009:
6005:
6002:
6000:
5997:
5992:
5988:
5985:
5979:
5975:
5972:
5969:
5966:
5963:
5960:
5957:
5954:
5951:
5948:
5944:
5940:
5937:
5934:
5931:
5928:
5925:
5922:
5921:
5918:
5915:
5912:
5909:
5905:
5897:
5893:
5888:
5884:
5881:
5877:
5871:
5866:
5862:
5858:
5855:
5852:
5849:
5846:
5843:
5840:
5838:
5835:
5829:
5825:
5820:
5816:
5807:
5803:
5799:
5796:
5791:
5787:
5783:
5780:
5777:
5772:
5767:
5763:
5758:
5754:
5751:
5748:
5747:
5724:
5721:
5718:
5715:
5710:
5706:
5700:
5695:
5691:
5687:
5684:
5681:
5678:
5675:
5672:
5661:
5660:
5649:
5646:
5643:
5640:
5637:
5634:
5631:
5628:
5625:
5622:
5619:
5616:
5611:
5607:
5603:
5600:
5595:
5591:
5587:
5584:
5581:
5576:
5572:
5568:
5565:
5562:
5559:
5556:
5553:
5508:
5505:
5492:
5491:
5480:
5477:
5474:
5471:
5468:
5463:
5459:
5453:
5450:
5447:
5443:
5439:
5436:
5433:
5430:
5427:
5424:
5419:
5415:
5409:
5406:
5403:
5399:
5395:
5392:
5389:
5386:
5383:
5380:
5361:restricted to
5359:
5358:
5345:
5339:
5336:
5330:
5325:
5322:
5265:
5262:
5254:
5253:
5242:
5239:
5236:
5233:
5230:
5225:
5221:
5217:
5214:
5210:
5206:
5203:
5200:
5197:
5192:
5188:
5182:
5177:
5173:
5167:
5163:
5159:
5156:
5153:
5148:
5144:
5138:
5134:
5130:
5127:
5124:
5121:
5118:
5114:
5110:
5107:
5104:
5101:
5091:
5087:
5083:
5079:
5075:
5072:
5066:
5062:
5054:
5049:
5045:
5036:
5032:
5026:
5021:
5017:
5010:
5002:
4998:
4994:
4990:
4986:
4983:
4979:
4973:
4968:
4964:
4955:
4951:
4945:
4941:
4937:
4934:
4931:
4928:
4925:
4922:
4919:
4916:
4902:
4901:
4886:
4882:
4878:
4875:
4872:
4869:
4859:
4855:
4851:
4847:
4843:
4840:
4834:
4830:
4822:
4817:
4813:
4809:
4806:
4804:
4802:
4798:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4775:
4772:
4768:
4764:
4761:
4758:
4755:
4751:
4743:
4739:
4735:
4731:
4727:
4724:
4720:
4714:
4709:
4705:
4701:
4698:
4696:
4694:
4691:
4687:
4683:
4680:
4677:
4674:
4667:
4663:
4659:
4655:
4651:
4644:
4640:
4636:
4632:
4626:
4622:
4618:
4615:
4612:
4609:
4606:
4603:
4600:
4596:
4592:
4589:
4586:
4583:
4576:
4572:
4568:
4564:
4556:
4552:
4548:
4544:
4538:
4535:
4532:
4528:
4524:
4521:
4517:
4513:
4510:
4507:
4504:
4497:
4493:
4489:
4485:
4479:
4474:
4470:
4466:
4463:
4460:
4457:
4455:
4453:
4447:
4442:
4438:
4435:
4430:
4426:
4420:
4416:
4411:
4405:
4402:
4397:
4393:
4389:
4386:
4383:
4380:
4377:
4374:
4368:
4363:
4359:
4356:
4351:
4347:
4341:
4337:
4332:
4326:
4323:
4320:
4317:
4313:
4309:
4306:
4301:
4297:
4293:
4290:
4287:
4284:
4281:
4278:
4275:
4273:
4268:
4258:
4254:
4245:
4240:
4234:
4231:
4228:
4227:
4204:
4200:
4196:
4193:
4190:
4187:
4182:
4178:
4172:
4167:
4163:
4159:
4156:
4153:
4150:
4147:
4144:
4121:
4117:
4114:
4110:
4107:
4104:
4101:
4081:
4077:
4073:
4070:
4067:
4062:
4058:
4054:
4051:
4045:
4040:
4035:
4031:
4025:
4017:
4013:
4009:
4006:
4002:
3998:
3995:
3992:
3987:
3984:
3981:
3977:
3956:
3951:
3947:
3943:
3938:
3934:
3930:
3927:
3923:
3919:
3916:
3913:
3908:
3904:
3881:
3876:
3872:
3866:
3861:
3857:
3853:
3850:
3845:
3840:
3836:
3830:
3825:
3821:
3817:
3813:
3809:
3806:
3784:
3780:
3775:
3772:
3768:
3763:
3759:
3738:
3735:
3732:
3729:
3709:
3674:
3650:
3647:
3644:
3641:
3618:
3592:
3565:
3542:
3538:
3534:
3531:
3528:
3525:
3499:
3472:
3451:
3448:
3445:
3442:
3422:
3410:
3407:
3405:
3402:
3401:
3400:
3399:
3398:
3387:Lars Hörmander
3328:
3298:
3269:If there is a
3267:
3248:
3241:Melrose (1993)
3215:
3202:
3199:
3194:
3190:
3186:
3183:
3180:
3175:
3171:
3167:
3162:
3158:
3154:
3149:
3145:
3141:
3138:
3122:
3112:
3109:
3087:-structures,
3045:)). The work (
3028:
3027:
3009:
3006:
2979:
2978:
2956:
2953:
2951:
2948:
2927:
2924:
2921:
2918:
2898:
2895:
2892:
2881:
2880:
2862:
2858:
2854:
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2847:
2844:
2841:
2838:
2826:
2823:
2820:
2817:
2807:
2802:
2800:
2797:
2794:
2791:
2788:
2786:
2781:
2777:
2771:
2766:
2761:
2754:
2750:
2746:
2744:
2740:
2736:
2732:
2729:
2726:
2723:
2720:
2710:
2707:
2704:
2701:
2688:
2683:
2681:
2678:
2675:
2672:
2669:
2667:
2661:
2638:
2635:
2632:
2629:
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2610:
2607:
2583:
2582:
2475:
2472:
2469:
2444:
2441:
2437:
2433:
2430:
2427:
2424:
2413:
2412:
2401:
2398:
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2383:
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2374:
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2360:
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2328:
2324:
2318:
2314:
2310:
2307:
2304:
2277:
2274:
2271:
2268:
2265:
2242:
2239:
2219:
2208:
2207:
2206:
2205:
2194:
2174:
2171:
2168:
2165:
2155:
2143:
2140:
2137:
2134:
2114:
2111:
2108:
2105:
2085:
2082:
2079:
2076:
2056:
2051:
2047:
2026:
2021:
2017:
1996:
1993:
1990:
1987:
1984:
1980:
1976:
1973:
1970:
1967:
1964:
1961:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1918:
1914:
1910:
1907:
1902:
1898:
1894:
1891:
1881:
1866:
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1839:
1835:
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1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1242:
1238:
1234:
1231:
1228:
1205:
1185:
1165:
1145:
1125:
1109:
1106:
1063:
1062:
1025:(solutions of
1002:
999:
947:
940:
871:
868:
865:
843:
838:
834:
830:
827:
824:
819:
814:
810:
806:
784:
780:
757:
752:
748:
744:
741:
738:
733:
728:
724:
686:
682:
659:
655:
651:
647:
643:
613:
609:
586:
582:
578:
574:
570:
561:and replacing
544:
540:
536:
533:
530:
525:
521:
517:
512:
508:
504:
501:
498:
493:
489:
458:
454:
450:
447:
444:
439:
435:
410:
407:
406:
405:
383:
376:vector bundles
365:
349:
346:
345:
344:
338:
332:
319:
306:
300:
294:
277:
274:quasiconformal
264:
246:
240:
234:
228:
214:
185:Richard Palais
177:Dirac operator
140:
137:
105:Isadore Singer
101:Michael Atiyah
87:
86:
69:
65:
64:
61:
60:First proof in
57:
56:
54:Isadore Singer
50:Michael Atiyah
47:
46:First proof by
43:
42:
37:
26:
9:
6:
4:
3:
2:
9151:
9140:
9137:
9135:
9132:
9130:
9127:
9126:
9124:
9114:
9110:
9106:
9099:
9094:
9093:
9083:
9079:
9075:
9071:
9068:
9064:
9063:
9058:
9053:
9048:
9042:
9023:
9018:
9017:
9002:
8998:
8994:
8990:
8985:
8979:
8975:
8971:
8967:
8964:
8960:
8955:
8950:
8946:
8942:
8938:
8934:
8930:
8926:
8922:
8919:
8915:
8911:
8907:
8903:
8899:
8894:
8891:
8887:
8882:
8877:
8873:
8869:
8864:
8861:
8857:
8853:
8849:
8845:
8841:
8837:
8833:
8829:
8824:
8821:
8817:
8813:
8809:
8805:
8801:
8797:
8793:
8788:
8785:
8781:
8777:
8773:
8769:
8765:
8761:
8757:
8753:
8749:
8745:
8742:
8738:
8734:
8728:
8724:
8720:
8716:
8712:
8708:
8704:
8701:
8695:
8691:
8687:
8682:
8677:
8673:
8668:
8664:
8658:
8654:
8653:
8648:
8644:
8640:
8636:
8635:
8627:
8623:
8622:Novikov, S.P.
8619:
8615:
8609:
8605:
8604:
8598:
8595:
8593:0-691-08542-0
8589:
8585:
8584:Spin Geometry
8581:
8577:
8573:
8569:
8565:
8561:
8557:
8553:
8548:
8544:
8540:
8536:
8532:
8528:
8525:
8521:
8517:
8513:
8509:
8505:
8500:
8497:
8493:
8489:
8485:
8481:
8477:
8472:
8467:
8463:
8459:
8456:-homologie",
8455:
8451:
8446:
8443:
8441:9780191589201
8437:
8433:
8428:
8423:
8418:
8413:
8408:
8403:
8398:
8394:
8388:
8385:, CRC Press,
8384:
8383:
8377:
8373:
8368:
8364:
8360:
8355:
8352:
8348:
8344:
8340:
8336:
8332:
8328:
8324:
8320:
8316:
8312:
8309:
8308:0-387-13619-3
8305:
8300:
8296:
8292:
8288:
8284:
8280:
8276:
8272:
8269:
8265:
8260:
8255:
8251:
8247:
8243:
8239:
8235:
8232:
8228:
8223:
8218:
8214:
8210:
8206:
8202:
8198:
8195:
8191:
8186:
8181:
8177:
8173:
8166:
8162:
8158:
8155:
8151:
8147:
8141:
8136:
8135:
8129:
8125:
8122:
8118:
8114:
8110:
8106:
8102:
8098:
8094:
8090:
8086:
8082:
8079:
8075:
8071:
8067:
8063:
8058:
8053:
8049:
8045:
8041:
8037:
8033:
8027:
8023:
8018:
8015:
8011:
8007:
8003:
7999:
7995:
7991:
7987:
7983:
7979:
7975:
7971:
7967:
7964:
7960:
7956:
7950:
7946:
7942:
7938:
7935:
7931:
7927:
7921:
7917:
7916:
7911:
7907:
7904:
7900:
7895:
7890:
7886:
7882:
7878:
7874:
7873:Invent. Math.
7867:
7863:
7859:
7855:
7851:
7847:
7843:
7839:
7835:
7831:
7830:Invent. Math.
7827:
7823:
7819:
7816:
7812:
7807:
7802:
7798:
7794:
7790:
7786:
7785:Invent. Math.
7779:
7775:
7771:
7767:
7763:
7759:
7755:
7751:
7747:
7743:
7742:Invent. Math.
7739:
7738:Patodi, V. K.
7735:
7731:
7727:
7723:
7719:
7715:
7711:
7707:
7703:
7702:
7694:
7690:
7686:
7682:
7678:
7674:
7670:
7666:
7665:Atiyah, M. F.
7662:
7657:
7652:
7649:(2): 245–50,
7648:
7644:
7640:
7636:
7635:Atiyah, M. F.
7632:
7628:
7624:
7620:
7616:
7612:
7608:
7607:
7602:
7598:
7594:
7590:
7586:
7582:
7578:
7574:
7570:
7569:
7564:
7560:
7556:
7552:
7548:
7544:
7540:
7536:
7532:
7531:
7526:
7522:
7518:
7514:
7510:
7506:
7502:
7498:
7494:
7490:
7486:
7482:
7477:
7472:
7468:
7464:
7460:
7456:
7452:
7448:
7444:
7440:
7436:
7432:
7428:
7424:
7420:
7419:Atiyah, M. F.
7416:
7413:
7409:
7405:
7401:
7400:Atiyah, M. F.
7397:
7394:
7390:
7386:
7382:
7381:Atiyah, M. F.
7378:
7377:
7375:
7373:
7369:
7353:
7351:0-7503-0606-8
7347:
7343:
7336:
7330:
7325:
7319:
7314:
7300:
7296:
7290:
7283:
7281:0-691-08542-0
7277:
7273:
7272:Spin Geometry
7269:
7265:
7259:
7252:
7246:
7242:
7238:
7233:
7228:
7224:
7217:
7210:
7205:
7198:
7193:
7186:
7181:
7174:
7169:
7162:
7157:
7150:
7145:
7138:
7133:
7126:
7121:
7114:
7113:Sullivan 1979
7109:
7102:
7097:
7090:
7085:
7078:
7077:Kasparov 1972
7073:
7066:
7061:
7054:
7049:
7042:
7037:
7030:
7025:
7018:
7013:
7006:
7002:
6998:
6994:
6989:
6982:
6977:
6970:
6965:
6958:
6953:
6946:
6945:Gel'fand 1960
6941:
6935:, p. 11.
6934:
6933:Hamilton 2020
6929:
6922:
6917:
6910:
6905:
6903:
6898:
6887:
6884:
6878:
6875:
6874:
6868:
6865:
6861:
6857:
6853:
6833:
6826:
6822:
6818:
6815:
6812:
6808:
6804:
6800:
6797:
6794:
6790:
6785:
6780:
6776:
6772:
6769:
6765:
6761:
6757:
6754:
6751:
6743:
6739:
6732:
6729:
6726:
6723:
6720:
6714:
6708:
6705:
6698:
6697:
6696:
6695:is given by
6694:
6690:
6686:
6681:
6677:
6673:
6669:
6664:
6662:
6658:
6654:
6650:
6649:heat equation
6646:
6642:
6638:
6631:Heat equation
6628:
6623:
6619:
6615:
6611:
6607:
6603:
6599:
6595:
6591:
6584:
6580:
6576:
6572:
6568:
6558:
6556:
6552:
6548:
6544:
6539:
6537:
6533:
6529:
6519:
6515:
6513:
6509:
6505:
6499:
6497:
6491:
6476:
6474:
6469:
6466:
6456:
6454:
6450:
6446:
6441:
6439:
6435:
6431:
6427:
6406:
6401:
6385:
6376:
6370:
6363:
6360:
6353:
6352:
6351:
6350:
6346:
6342:
6321:
6318:
6315:
6309:
6305:
6296:
6292:
6288:
6284:
6279:
6277:
6273:
6269:
6265:
6261:
6251:
6249:
6245:
6223:
6220:
6214:
6211:
6205:
6199:
6196:
6191:
6187:
6183:
6177:
6174:
6171:
6165:
6158:
6157:
6156:
6154:
6128:
6125:
6112:
6108:
6103:
6099:
6096:
6089:
6085:
6081:
6073:
6068:
6064:
6053:
6049:
6045:
6041:
6037:
6034:
6028:
6024:
6016:
6011:
6007:
6003:
6001:
5995:
5986:
5983:
5977:
5973:
5970:
5964:
5961:
5955:
5952:
5949:
5938:
5935:
5932:
5926:
5923:
5913:
5910:
5903:
5895:
5891:
5886:
5882:
5879:
5875:
5869:
5864:
5860:
5853:
5847:
5844:
5841:
5839:
5833:
5823:
5818:
5814:
5805:
5797:
5794:
5789:
5781:
5778:
5770:
5765:
5761:
5756:
5752:
5749:
5738:
5737:
5736:
5719:
5716:
5708:
5704:
5698:
5693:
5689:
5685:
5679:
5676:
5670:
5644:
5641:
5638:
5632:
5629:
5623:
5620:
5617:
5609:
5605:
5601:
5598:
5593:
5585:
5582:
5574:
5570:
5566:
5560:
5554:
5551:
5544:
5543:
5542:
5540:
5536:
5532:
5528:
5524:
5478:
5469:
5461:
5457:
5451:
5448:
5445:
5437:
5434:
5425:
5417:
5413:
5407:
5404:
5401:
5393:
5390:
5384:
5378:
5371:
5370:
5369:
5366:
5364:
5343:
5328:
5311:
5310:
5309:
5307:
5303:
5299:
5295:
5291:
5287:
5283:
5279:
5275:
5271:
5261:
5259:
5237:
5234:
5228:
5223:
5219:
5215:
5204:
5201:
5198:
5190:
5186:
5180:
5175:
5171:
5165:
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5136:
5128:
5125:
5119:
5108:
5105:
5102:
5089:
5085:
5081:
5077:
5073:
5070:
5064:
5060:
5052:
5047:
5043:
5034:
5030:
5024:
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5008:
5000:
4996:
4992:
4988:
4984:
4981:
4977:
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4966:
4962:
4953:
4949:
4943:
4935:
4932:
4926:
4920:
4914:
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4906:
4905:
4876:
4873:
4870:
4857:
4853:
4849:
4845:
4841:
4838:
4832:
4828:
4820:
4815:
4811:
4807:
4805:
4792:
4789:
4786:
4780:
4777:
4762:
4759:
4756:
4749:
4741:
4737:
4733:
4729:
4725:
4722:
4718:
4712:
4707:
4703:
4699:
4697:
4681:
4678:
4675:
4665:
4661:
4657:
4653:
4649:
4642:
4638:
4634:
4630:
4624:
4616:
4613:
4607:
4604:
4601:
4590:
4587:
4584:
4574:
4570:
4566:
4562:
4554:
4550:
4546:
4542:
4536:
4533:
4530:
4526:
4522:
4511:
4508:
4505:
4495:
4491:
4487:
4483:
4477:
4472:
4468:
4464:
4461:
4458:
4456:
4445:
4436:
4433:
4428:
4424:
4418:
4409:
4403:
4400:
4395:
4387:
4384:
4378:
4375:
4372:
4366:
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4354:
4349:
4345:
4339:
4330:
4324:
4321:
4318:
4307:
4304:
4299:
4295:
4288:
4285:
4282:
4279:
4276:
4274:
4266:
4252:
4238:
4232:
4229:
4218:
4217:
4216:
4194:
4191:
4188:
4180:
4176:
4170:
4165:
4161:
4157:
4151:
4148:
4142:
4133:
4119:
4115:
4112:
4108:
4105:
4102:
4099:
4071:
4068:
4060:
4056:
4052:
4049:
4043:
4033:
4029:
4023:
4015:
4011:
4007:
3996:
3993:
3985:
3982:
3979:
3975:
3949:
3945:
3936:
3932:
3928:
3917:
3914:
3906:
3902:
3874:
3870:
3864:
3859:
3855:
3851:
3848:
3838:
3834:
3828:
3823:
3819:
3815:
3807:
3804:
3782:
3778:
3773:
3770:
3766:
3761:
3757:
3736:
3733:
3730:
3727:
3707:
3699:
3694:
3692:
3688:
3672:
3664:
3645:
3639:
3632:
3616:
3540:
3536:
3532:
3529:
3526:
3523:
3462:. If we take
3449:
3446:
3443:
3440:
3420:
3413:Suppose that
3396:
3392:
3388:
3384:
3383:Boris Fedosov
3380:
3376:
3375:
3373:
3369:
3365:
3361:
3357:
3353:
3349:
3345:
3344:axial anomaly
3341:
3337:
3333:
3329:
3326:
3322:
3318:
3314:
3313:index theorem
3312:
3307:
3302:
3301:Atiyah (1976)
3299:
3296:
3292:
3288:
3284:
3280:
3276:
3272:
3268:
3265:
3261:
3257:
3253:
3249:
3246:
3242:
3237:
3233:
3229:
3225:
3220:
3216:
3200:
3192:
3188:
3181:
3173:
3169:
3160:
3156:
3147:
3143:
3136:
3129:
3128:
3123:
3119:
3115:
3114:
3108:
3106:
3102:
3098:
3094:
3090:
3086:
3082:
3077:
3075:
3074:Sullivan 1979
3069:
3067:
3063:
3059:
3055:
3050:
3048:
3044:
3039:
3037:
3033:
3026:
3023:
3022:
3021:
3019:
3015:
3005:
3002:
3000:
2996:
2995:Kasparov 1972
2992:
2988:
2984:
2977:
2975:
2970:
2969:
2968:
2966:
2962:
2947:
2945:
2941:
2922:
2916:
2896:
2893:
2890:
2848:
2845:
2839:
2824:
2818:
2805:
2795:
2789:
2779:
2775:
2752:
2748:
2730:
2727:
2721:
2708:
2702:
2686:
2676:
2670:
2652:
2651:
2650:
2636:
2630:
2627:
2624:
2616:
2606:
2604:
2600:
2595:
2591:
2589:
2581:
2579:
2574:
2573:
2572:
2570:
2566:
2562:
2558:
2557:index problem
2553:
2551:
2547:
2543:
2539:
2534:
2532:
2528:
2524:
2520:
2516:
2512:
2508:
2504:
2500:
2496:
2492:
2487:
2473:
2470:
2467:
2460:
2442:
2439:
2431:
2428:
2422:
2396:
2390:
2387:
2378:
2375:
2369:
2361:
2355:
2352:
2349:
2343:
2337:
2334:
2326:
2322:
2316:
2308:
2305:
2295:
2294:
2293:
2291:
2272:
2269:
2263:
2256:
2240:
2237:
2217:
2192:
2169:
2163:
2156:
2138:
2132:
2109:
2103:
2080:
2074:
2054:
2049:
2045:
2024:
2019:
2015:
1988:
1982:
1978:
1971:
1965:
1959:
1933:
1927:
1924:
1921:
1916:
1912:
1908:
1905:
1900:
1896:
1889:
1882:
1880:
1856:
1853:
1845:
1841:
1829:
1823:
1817:
1814:
1811:
1804:
1790:
1781:
1775:
1771:
1764:
1758:
1755:
1752:
1744:
1720:
1714:
1708:
1704:
1697:
1691:
1683:
1680:
1677:
1673:
1658:
1655:
1647:
1643:
1639:
1636:
1629:
1628:
1602:
1596:
1593:
1590:
1585:
1581:
1577:
1574:
1569:
1565:
1558:
1552:
1549:
1541:
1538:
1534:
1510:
1504:
1501:
1494:
1479:
1471:
1452:
1446:
1443:
1436:
1435:
1434:
1420:
1412:
1393:
1387:
1384:
1378:
1372:
1369:
1362:
1340:
1334:
1331:
1325:
1319:
1316:
1311:
1307:
1301:
1293:
1290:
1284:
1278:
1269:
1263:
1260:
1254:
1248:
1245:
1240:
1232:
1229:
1219:
1218:
1217:
1203:
1183:
1163:
1143:
1123:
1115:
1105:
1102:
1098:
1094:
1090:
1086:
1082:
1078:
1074:
1072:
1068:
1060:
1056:
1052:
1048:
1047:
1046:
1044:
1040:
1036:
1032:
1028:
1024:
1020:
1016:
1012:
1008:
998:
996:
991:
987:
983:
980:′ such that
979:
975:
974:pseudoinverse
971:
970:
965:
960:
958:
954:
950:
943:
936:
932:
928:
924:
920:
916:
912:
908:
904:
900:
896:
892:
887:
885:
869:
866:
863:
841:
836:
832:
828:
825:
822:
817:
812:
808:
804:
782:
778:
755:
750:
746:
742:
739:
736:
731:
726:
722:
713:
708:
706:
702:
684:
680:
657:
653:
645:
633:
629:
611:
607:
584:
580:
572:
560:
542:
538:
534:
531:
528:
523:
519:
515:
510:
506:
502:
499:
496:
491:
487:
478:
474:
456:
452:
448:
445:
442:
437:
433:
424:
420:
416:
403:
399:
395:
391:
387:
384:
381:
377:
373:
369:
366:
363:
359:
355:
352:
351:
342:
339:
336:
333:
330:
326:
323:
320:
317:
313:
310:
307:
304:
301:
298:
295:
292:
288:
284:
281:
278:
275:
271:
268:
265:
262:
261:heat equation
258:
254:
250:
247:
244:
241:
238:
235:
232:
229:
226:
222:
218:
215:
212:
208:
205:
202:
201:
200:
198:
194:
190:
186:
180:
178:
174:
170:
166:
162:
158:
154:
150:
146:
136:
134:
130:
126:
122:
118:
114:
110:
106:
102:
98:
94:
85:
81:
77:
73:
70:
66:
62:
58:
55:
51:
48:
44:
41:
38:
34:
19:
9104:
9082:the original
9060:
9029:. Retrieved
8973:
8944:
8940:
8928:
8901:
8897:
8871:
8867:
8835:
8831:
8795:
8791:
8759:
8755:
8748:Sullivan, D.
8722:
8719:Sullivan, D.
8710:
8707:Singer, I.M.
8671:
8651:
8638:
8632:
8602:
8583:
8567:
8542:
8538:
8507:
8503:
8471:math/9905210
8461:
8457:
8453:
8449:
8431:
8381:
8362:
8358:
8326:
8322:
8282:
8278:
8249:
8245:
8242:Sullivan, D.
8212:
8208:
8205:Sullivan, D.
8175:
8171:
8133:
8096:
8092:
8069:
8047:
8043:
8021:
7989:
7985:
7944:
7914:
7876:
7872:
7833:
7829:
7788:
7784:
7745:
7741:
7705:
7699:
7676:
7672:
7646:
7642:
7610:
7604:
7572:
7566:
7534:
7528:
7496:
7492:
7466:
7462:
7430:
7426:
7423:Segal, G. B.
7403:
7384:
7365:
7341:
7335:
7324:
7313:
7302:. Retrieved
7298:
7289:
7271:
7258:
7222:
7216:
7204:
7192:
7180:
7168:
7161:Teleman 1984
7156:
7149:Teleman 1983
7144:
7132:
7125:Getzler 1983
7120:
7108:
7101:Melrose 1993
7096:
7084:
7072:
7060:
7048:
7036:
7024:
7017:Novikov 1965
7012:
6988:
6976:
6964:
6952:
6940:
6928:
6916:
6886:Witten index
6863:
6859:
6855:
6851:
6849:
6692:
6688:
6684:
6679:
6675:
6671:
6667:
6665:
6657:Melrose 1993
6635:Atiyah,
6634:
6621:
6617:
6613:
6609:
6605:
6601:
6597:
6593:
6589:
6582:
6578:
6574:
6570:
6564:
6554:
6550:
6546:
6542:
6540:
6525:
6516:
6511:
6507:
6503:
6500:
6493:
6470:
6462:
6452:
6448:
6444:
6442:
6437:
6429:
6425:
6423:
6344:
6294:
6290:
6286:
6282:
6281:The bundles
6280:
6267:
6263:
6257:
6247:
6243:
6241:
6150:
5662:
5538:
5530:
5526:
5522:
5493:
5367:
5362:
5360:
5305:
5301:
5297:
5296:of type (0,
5293:
5289:
5285:
5281:
5277:
5269:
5267:
5255:
4903:
4134:
3695:
3412:
3381:proposed by
3371:
3367:
3366:) over the (
3363:
3359:
3355:
3340:Roman Jackiw
3310:
3309:
3293:. See also:
3290:
3278:
3274:
3271:group action
3263:
3259:
3255:
3251:
3125:
3104:
3096:
3092:
3088:
3084:
3078:
3070:
3062:Teleman 1985
3053:
3051:
3040:
3031:
3029:
3024:
3011:
3003:
2991:Teleman 1983
2987:Teleman 1983
2983:Teleman 1980
2980:
2971:
2965:Teleman 1984
2961:Teleman 1983
2958:
2882:
2612:
2602:
2598:
2596:
2592:
2587:
2584:
2577:
2575:
2568:
2564:
2560:
2556:
2554:
2549:
2545:
2541:
2537:
2535:
2530:
2526:
2522:
2518:
2514:
2510:
2506:
2502:
2498:
2494:
2490:
2488:
2414:
2209:
1526:is equal to
1358:
1216:is given by
1113:
1111:
1100:
1096:
1092:
1088:
1084:
1080:
1076:
1075:
1070:
1066:
1064:
1058:
1054:
1050:
1042:
1038:
1034:
1026:
1022:
1014:
1006:
1004:
989:
985:
981:
977:
973:
967:
963:
961:
956:
952:
945:
938:
934:
930:
926:
922:
918:
914:
906:
902:
894:
890:
888:
883:
711:
709:
707:is nonzero.
704:
700:
631:
630:, of degree
627:
558:
476:
422:
418:
414:
412:
401:
397:
393:
389:
385:
379:
371:
367:
353:
340:
334:
328:
321:
312:Alain Connes
308:
302:
296:
283:Ezra Getzler
279:
266:
257:Vijay Patodi
248:
242:
236:
230:
217:Robion Kirby
203:
181:
169:Armand Borel
142:
120:
116:
99:, proved by
96:
90:
68:Consequences
18:Index theory
8874:: 117–152,
8838:: 251–290,
8365:: 111–117,
8315:Getzler, E.
8252:: 181–252,
8099:: 257–360,
7992:: 155–191,
7173:Connes 1986
7137:Witten 1982
7065:Singer 1971
7053:Atiyah 1970
6957:Palais 1965
6921:Kayani 2020
6661:Gilkey 1994
6651:, see e.g.
5521:. Then the
5308:be the sum
3687:Euler class
3352:Higgs field
3350:called the
3342:, used the
3273:of a group
3101:Hilsum 1999
3058:Singer 1971
2974:Atiyah 1970
2569:topological
2255:Euler class
471:, then its
374:are smooth
9123:Categories
9031:January 3,
8997:Hirzebruch
8890:0547.58036
8860:0531.58044
8784:0531.58045
8741:0478.57007
8422:1910.01080
8407:1512.02632
8268:0704.57008
8231:0840.57013
8201:Connes, A.
8194:0759.58047
8161:Connes, A.
8154:0818.46076
8128:Connes, A.
8121:0592.46056
8085:Connes, A.
8078:0149.41102
8014:0332.14003
7974:Fulton, W.
7730:Atiyah, M.
7393:0193.43601
7362:References
7304:2021-02-05
6339:times the
6297:-forms as
6244:projective
3797:such that
3391:Raoul Bott
2588:difference
2536:As usual,
1470:Todd class
969:parametrix
911:jet bundle
479:variables
425:variables
253:Raoul Bott
159:, and the
9067:EMS Press
8918:121137053
8852:121497293
8820:122247909
8676:CiteSeerX
8641:: 298–300
8560:Kirby, R.
8531:Kirby, R.
8496:119708566
8113:122740195
8050:: 56–99,
7943:(1988b),
7912:(1988a),
7866:189831012
7778:115700319
7227:CiteSeerX
7041:Thom 1956
6893:Citations
6827:∗
6813:−
6801:
6795:−
6781:∗
6770:−
6758:
6744:∗
6733:
6727:
6709:
6522:Cobordism
6395:∗
6367:Δ
6364:≡
6319:−
6215:
6200:
6188:∫
6166:χ
6100:−
6082:−
6065:∏
6046:−
6038:−
6008:∏
5991:¯
5974:
5956:
5939:⊗
5927:
5883:−
5861:∏
5848:
5828:¯
5819:∗
5802:Λ
5798:⊗
5779:−
5762:∑
5753:
5690:∏
5633:χ
5602:
5583:−
5571:∑
5555:
5507:¯
5504:∂
5479:⋯
5476:→
5462:∗
5442:Λ
5438:⊗
5432:→
5418:∗
5398:Λ
5394:⊗
5388:→
5382:→
5344:∗
5338:¯
5335:∂
5324:¯
5321:∂
5220:∫
5205:⊗
5172:∏
5155:−
5143:∫
5126:−
5109:⊗
5082:−
5074:−
5044:∏
5016:∏
4993:−
4985:−
4963:∏
4950:∫
4933:−
4915:χ
4877:⊗
4850:−
4842:−
4812:∏
4793:⊗
4781:
4763:⊗
4734:−
4726:−
4704:∏
4682:⊗
4658:−
4650:⋯
4635:−
4614:−
4605:…
4591:⊗
4567:−
4547:−
4527:∑
4512:⊗
4488:−
4469:∑
4465:−
4437:⊗
4429:∗
4415:Λ
4404:
4385:−
4376:…
4373:−
4358:⊗
4350:∗
4336:Λ
4325:
4308:⊗
4300:∗
4289:
4283:−
4257:Λ
4253:−
4244:Λ
4233:
4195:⊗
4162:∏
4113:…
4072:⊗
4053:−
4039:¯
3997:⊗
3918:⊗
3880:¯
3865:⊕
3852:⋯
3849:⊕
3844:¯
3829:⊕
3808:⊗
3771:…
3640:χ
3591:Λ
3564:Λ
3541:∗
3498:Λ
3471:Λ
3198:→
3185:→
3182:⋯
3179:→
3166:→
3153:→
3140:→
3066:Thom 1956
2959:Due to (
2944:Chow ring
2897:∗
2825:⋅
2780:∗
2753:∗
2709:⋅
2634:→
2440:−
2391:
2356:
2350:−
2338:
2323:∫
2306:−
2164:σ
2104:σ
2050:∗
2020:∗
1928:σ
1917:∗
1901:∗
1846:∗
1838:→
1830:⊗
1788:→
1670:→
1637:φ
1597:σ
1586:∗
1570:∗
1553:
1539:−
1535:φ
1505:
1447:
1388:
1373:
1335:
1320:
1308:∫
1291:−
1264:
1249:
1230:−
867:≥
826:⋯
805:−
740:⋯
650:∂
642:∂
577:∂
569:∂
532:…
500:…
446:…
225:René Thom
193:cobordism
9041:cite web
8925:Thom, R.
8649:(1965),
8624:(1965),
8582:(1989),
8566:(1977),
8359:Topology
8351:55438589
8317:(1983),
8209:Topology
8172:Topology
8130:(1994),
8087:(1986),
8068:(1965),
8006:83458307
7980:(1979),
7970:Baum, P.
7836:: 1–62,
7734:Bott, R.
7669:Bott, R.
7639:Bott, R.
7270:(1989),
6871:See also
6567:K-theory
6561:K-theory
6549:) where
6428:, where
6347:is the
5272:to be a
3404:Examples
3219:manifold
3012:Due to (
2806:→
2770:↓
2760:↓
2687:→
1627:, where
1077:Example:
1031:cokernel
935:elliptic
701:elliptic
362:manifold
348:Notation
251:Atiyah,
197:K-theory
8963:0683171
8800:Bibcode
8776:8348213
8512:Bibcode
8331:Bibcode
8287:Bibcode
7963:0951895
7934:0951894
7903:0550183
7881:Bibcode
7858:0463358
7838:Bibcode
7815:0650829
7793:Bibcode
7770:0650828
7750:Bibcode
7722:1970721
7693:1970694
7627:1970757
7589:1970756
7551:1970717
7513:1970715
7447:1970716
7412:0420729
6674:, then
6659:) and (
6643: (
6465:Â genus
6272:L genus
5300:) with
3661:of the
3629:is the
3243:of the
3217:If the
3121:easier.
2938:is the
2501:) to K(
1877:is the
1741:is the
1468:is the
1409:on the
1087:), and
982:DD′ -1
360:smooth
358:compact
173:Â genus
139:History
9001:Singer
8989:Atiyah
8980:
8961:
8916:
8888:
8858:
8850:
8818:
8782:
8774:
8739:
8729:
8696:
8678:
8659:
8610:
8590:
8494:
8488:121079
8486:
8438:
8389:
8349:
8306:
8266:
8229:
8192:
8152:
8142:
8119:
8111:
8076:
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8466:arXiv
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