14824:
13570:
4312:
7494:
4928:
14607:
4274:
14845:
6300:
14813:
12393:
14882:
14855:
14835:
12277:
7291:
12353:
12315:
12595:
6101:
4890:
12851:, which implies that it has trivial homology. Consequently, additional cuts disconnect it. The square is not the only shape in the plane that can be glued into a surface. Gluing opposite sides of an octagon, for example, produces a surface with two holes. In fact, all closed surfaces can be produced by gluing the sides of some polygon and all even-sided polygons (2
432:
7489:{\displaystyle \dotsb {\overset {\partial _{n+1}}{\longrightarrow \,}}C_{n}{\overset {\partial _{n}}{\longrightarrow \,}}C_{n-1}{\overset {\partial _{n-1}}{\longrightarrow \,}}\dotsb {\overset {\partial _{2}}{\longrightarrow \,}}C_{1}{\overset {\partial _{1}}{\longrightarrow \,}}C_{0}{\overset {\epsilon }{\longrightarrow \,}}\mathbb {Z} {\longrightarrow \,}0}
6295:{\displaystyle \dotsb {\overset {\partial _{n+1}}{\longrightarrow \,}}C_{n}{\overset {\partial _{n}}{\longrightarrow \,}}C_{n-1}{\overset {\partial _{n-1}}{\longrightarrow \,}}\dotsb {\overset {\partial _{2}}{\longrightarrow \,}}C_{1}{\overset {\partial _{1}}{\longrightarrow \,}}C_{0}{\overset {\partial _{0}}{\longrightarrow \,}}0}
13523:. Homology groups are finitely generated abelian groups, and homology classes are elements of these groups. The Betti numbers of the manifold are the rank of the free part of the homology group, and in the special case of surfaces, the torsion part of the homology group only occurs for non-orientable cycles.
12082:. By linking nearest neighbor points in the cloud into a triangulation, a simplicial approximation of the manifold is created and its simplicial homology may be calculated. Finding techniques to robustly calculate homology using various triangulation strategies over multiple length scales is the topic of
1027:
to other types of mathematical objects, one first gives a prescription for associating chain complexes to that object, and then takes the homology of such a chain complex. For the homology theory to be valid, all such chain complexes associated to the same mathematical object must have the same
9210:
The different types of homology theory arise from functors mapping from various categories of mathematical objects to the category of chain complexes. In each case the composition of the functor from objects to chain complexes and the functor from chain complexes to homology groups defines the
5394:
236:
12590:
The edges of the square may then be glued back together in different ways. The square can be twisted to allow edges to meet in the opposite direction, as shown by the arrows in the diagram. The various ways of gluing the sides yield just four topologically distinct surfaces:
5617:
5825:
11563:
9779:
4880:
4739:
4488:
5200:
8962:, which is not commutative: looping around the lefthand cycle and then around the righthand cycle is different from looping around the righthand cycle and then looping around the lefthand cycle. By contrast, the figure eight's first homology group
12844:, since this cycle lives in a torsion-free homology class. This corresponds to the fact that in the fundamental polygon of the Klein bottle, only one pair of sides is glued with a twist, whereas in the projective plane both sides are twisted.
7213:
10084:
5268:
12093:, sensors may communicate information via an ad-hoc network that dynamically changes in time. To understand the global context of this set of local measurements and communication paths, it is useful to compute the homology of the
2583:
have no boundary, one can associate cycles to each of these spaces. However, the chain complex notion of cycles (elements whose boundary is a "zero chain") is more general than the topological notion of a shape with no boundary.
1063:
of the mathematical object being studied to the category of abelian groups and group homomorphisms, or more generally to the category corresponding to the associated chain complexes. One can also formulate homology theories as
5265:. The torus has a single path-connected component, two independent one-dimensional holes (indicated by circles in red and blue) and one two-dimensional hole as the interior of the torus. The corresponding homology groups are
7603:
1790:
873:
427:{\displaystyle C_{\bullet }:\cdots \longrightarrow C_{n+1}{\stackrel {d_{n+1}}{\longrightarrow }}C_{n}{\stackrel {d_{n}}{\longrightarrow }}C_{n-1}{\stackrel {d_{n-1}}{\longrightarrow }}\cdots ,\quad d_{n}\circ d_{n+1}=0.}
1112:(see below) of that topological space, with the consequence that one often simply refers to the "homology" of that space, instead of specifying which homology theory was used to compute the homology groups in question.
6972:
6573:
9200:
5495:
1091:
Regardless of how they are formulated, homology theories help provide information about the structure of the mathematical objects to which they are associated, and can sometimes help distinguish different objects.
5713:
11349:
2419:
701:
6427:
12721:
were only wound once, it would also be non-shrinkable and reverse left and right. However it is wound a second time, which swaps right and left back again; it can be shrunk to a point and is homologous to
1659:
Since such constructions are somewhat technical, informal discussions of homology sometimes focus instead on topological notions that parallel some of the group-theoretic aspects of cycles and boundaries.
9012:
4780:
4639:
4388:
9924:
2587:
It is this topological notion of no boundary that people generally have in mind when they claim that cycles can intuitively be thought of as detecting holes. The idea is that for no-boundary shapes like
12632:, which is a torus with a twist in it (In the square diagram, the twist can be seen as the reversal of the bottom arrow). It is a theorem that the re-glued surface must self-intersect (when immersed in
11269:
11179:
5106:
8960:
8197:
9597:
9423:
2286:
10384:
6736:
5923:
is a new phenomenon: intuitively, it corresponds to the fact that there is a single non-contractible "loop", but if we do the loop twice, it becomes contractible to zero. This phenomenon is called
11364:
5066:
4593:
10915:
4132:
9611:
9358:
8122:
5666:
6089:
7019:
6620:
8699:
8572:
6801:
12027:
11992:
8297:
3658:
1441:
156:
6024:
5921:
5015:
is a solid disc. It has a single path-connected component, but in contrast to the circle, has no higher-dimensional holes. The corresponding homology groups are all trivial except for
11829:
5263:
11630:
11088:
10790:
10731:
7649:
1849:
1524:
5870:
2006:
10843:
3804:
1634:
7731:
7062:
6900:
6854:
6663:
4977:
2169:
12886:
of the manifold (Betti numbers are a refinement of the Euler characteristic). Classifying the non-orientable cycles requires additional information about torsion coefficients.
10287:
7847:
5488:
3325:
3293:
540:
12155:
of the solution based on the chosen boundary conditions and the homology of the domain. FEM domains can be triangulated, from which the simplicial homology can be calculated.
11788:
5704:
4636:
has a single connected component, no one-dimensional-boundary holes, a two-dimensional-boundary hole, and no higher-dimensional holes. The corresponding homology groups are
9499:
6336:
7815:
7679:
4510:
4164:
12598:
The four ways of gluing a square to make a closed surface: glue single arrows together and glue double arrows together so that the arrowheads point in the same direction.
8904:
8807:
8735:
8477:
8351:
2554:
11708:
10534:
9848:
2514:
2472:
7517:
1135:, which allows more general maps of simplices into the topological space. Replacing simplices with disks of various dimensions results in a related construction called
9088:
8771:
7926:
7890:
7793:
7767:
6499:
4252:
4200:
4000:
3964:
3900:
3737:
1362:
11924:
11746:
9452:
9283:
6463:
11871:
10967:
10597:
8598:
8051:
7073:
5100:
4775:
2093:
13489:
13205:
13149:
13093:
13037:
12980:
12695:
12626:
12542:
12505:
12459:
12423:
12385:
12345:
12307:
11292:
10469:
10438:
10411:
10314:
10234:
9951:
8418:
5431:
5013:
4920:
4634:
4536:
4379:
4342:
4304:
3584:
3553:
3506:
3459:
3250:
3223:
3192:
3018:
2991:
2964:
2937:
2910:
2883:
2856:
2829:
2802:
2775:
2748:
2721:
2694:
2667:
2640:
2613:
2581:
2196:
2064:
1931:
1904:
1578:
1551:
1314:
1287:
1249:
1222:
1195:
1005:
954:
900:
751:
594:
482:
220:
186:
12056:
9981:
9808:
8840:
8624:
9472:
9052:
9032:
8860:
8497:
8438:
8391:
8371:
4020:
3928:
3864:
3844:
3824:
3698:
3678:
3612:
3526:
3479:
3432:
3412:
3392:
3372:
3352:
3165:
3145:
3125:
3105:
3085:
3065:
3045:
2439:
2306:
1869:
1654:
1461:
1382:
974:
927:
721:
564:
455:
13497:
In a search for increased rigour, PoincarΓ© went on to develop the simplicial homology of a triangulated manifold and to create what is now called a simplicial
2038:
1966:
5935:
The following text describes a general algorithm for constructing the homology groups. It may be easier for the reader to look at some simple examples first:
4385:. It has a single connected component and a one-dimensional-boundary hole, but no higher-dimensional holes. The corresponding homology groups are given as
2516:. The closest topological analog of this idea would be a shape that has "no boundary," in the sense that its boundary is the empty set. For example, since
14885:
7928:
groups is usually rather difficult since they have a very large number of generators. On the other hand, there are tools which make the task easier.
7525:
5389:{\displaystyle H_{k}(T^{2})={\begin{cases}\mathbb {Z} &k=0,2\\\mathbb {Z} \times \mathbb {Z} &k=1\\\{0\}&{\text{otherwise}}\end{cases}}}
1677:
1044:
are respectively obtained from singular chain complexes, Morse complexes, Khovanov complexes, and
Hochschild complexes. In other cases, such as for
760:
3586:
into one of the 2 lobes of the figure-eight shape gives a boundary, despite the fact that it is possible to glue a disk onto a figure-eight lobe.
6905:
6506:
1879:, but it is also a notion familiar from examples, e.g., the boundary of the unit disk is the unit circle, or more topologically, the boundary of
9093:
80:. (This latter notion of homology admits more intuitive descriptions for 1- or 2-dimensional topological spaces, and is sometimes referenced in
13613:
12660:
is made, it returns on the other side and goes round the surface a second time before returning to its starting point, cutting out a twisted
12882:. 1β121 (1895). The paper introduced homology classes and relations. The possible configurations of orientable cycles are classified by the
12579:, it can be opened out and flattened into a rectangle or, more conveniently, a square. One opposite pair of sides represents the cut along
11310:
8906:
is the group of homotopy classes of directed loops starting and ending at a predetermined point (e.g. its center). It is isomorphic to the
2669:, it is possible in each case to glue on a larger shape for which the original shape is the boundary. For instance, starting with a circle
14170:
8813:
are poorly understood and are not known in general, in contrast to the straightforward description given above for the homology groups.
4051:
1320:
which refers to adding cycles symbolically rather than combining them geometrically. Any formal sum of cycles is again called a cycle.
2321:
603:
11956:
to the same point. (Two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.)
6352:
1328:
Explicit constructions of homology groups are somewhat technical. As mentioned above, an explicit realization of the homology groups
12828:
must be followed around twice to achieve a zero cycle, the surface is said to have a torsion coefficient of 2. However, following a
8965:
5612:{\displaystyle H_{k}(T^{n})={\begin{cases}\mathbb {Z} ^{\binom {n}{k}}&0\leq k\leq n\\\{0\}&{\text{otherwise}}\end{cases}}}
11198:
11108:
9853:
13734:
12664:. Because local left and right can be arbitrarily re-oriented in this way, the surface as a whole is said to be non-orientable.
8913:
8131:
14272:
5820:{\displaystyle H_{k}(P)={\begin{cases}\mathbb {Z} &k=0\\\mathbb {Z} _{2}&k=1\\\{0\}&{\text{otherwise}}\end{cases}}}
1463:, where the type of chain complex depends on the choice of homology theory in use. These cycles and boundaries are elements of
12701:, is visually complex, so a hemispherical embedding is shown in the diagram, in which antipodal points around the rim such as
11558:{\displaystyle \cdots \to H_{n}(A)\to H_{n}(B)\to H_{n}(C)\to H_{n-1}(A)\to H_{n-1}(B)\to H_{n-1}(C)\to H_{n-2}(A)\to \cdots }
9525:
9363:
2201:
1131:
to a one-dimensional simplex, and a triangle-based pyramid is a 3-simplex.) Simplicial homology can in turn be generalized to
14430:
14396:
14057:
11097:
are zero, and the others are finitely generated abelian groups (or finite-dimensional vector spaces), then we can define the
10319:
6672:
14148:
5632:
The two independent 1-dimensional holes form independent generators in a finitely-generated abelian group, expressed as the
11640:. This lemma can be applied to homology in numerous ways that aid in calculating homology groups, such as the theories of
1104:. For sufficiently nice topological spaces and compatible choices of coefficient rings, any homology theory satisfying the
5018:
4545:
13526:
The subsequent spread of homology groups brought a change of terminology and viewpoint from "combinatorial topology" to "
11192:
in the case of vector spaces). It turns out that the Euler characteristic can also be computed on the level of homology:
10848:
9774:{\displaystyle \partial _{n}(\sigma )=\sum _{i=0}^{n}(-1)^{i}\left(\sigma ,\dots ,\sigma ,\sigma ,\dots ,\sigma \right),}
12772:-cycles. Since the Klein bottle is nonorientable, you can transport one of them all the way round the bottle (along the
6342:< 0. It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all
1166:
One of the ideas that led to the development of homology was the observation that certain low-dimensional shapes can be
14519:
13694:
12544:
has closed curves which cannot be continuously deformed into each other, for example in the diagram none of the cycles
3260:
representatives of 1-cycles, 3-cycles, and oriented 2-cycles all admit manifold-shaped holes, but for example the real
1028:
homology. The resulting homology theory is often named according to the type of chain complex prescribed. For example,
12163:
Various software packages have been developed for the purposes of computing homology groups of finite cell complexes.
9292:
8064:
5638:
4875:{\displaystyle H_{k}\left(S^{n}\right)={\begin{cases}\mathbb {Z} &k=0,n\\\{0\}&{\text{otherwise}}\end{cases}}}
4734:{\displaystyle H_{k}\left(S^{2}\right)={\begin{cases}\mathbb {Z} &k=0,2\\\{0\}&{\text{otherwise}}\end{cases}}}
4483:{\displaystyle H_{k}\left(S^{1}\right)={\begin{cases}\mathbb {Z} &k=0,1\\\{0\}&{\text{otherwise}}\end{cases}}}
14873:
14868:
14463:
14366:
14303:
14282:
14246:
14219:
6032:
6977:
6578:
5195:{\displaystyle H_{k}\left(B^{n}\right)={\begin{cases}\mathbb {Z} &k=0\\\{0\}&{\text{otherwise}}\end{cases}}}
875:. One can optionally endow chain complexes with additional structure, for example by additionally taking the groups
8633:
8506:
7243:. Each homology class is an equivalence class over cycles and two cycles in the same homology class are said to be
4596:
13603:
6745:
14863:
11997:
11962:
9014:
is abelian. To express this explicitly in terms of homology classes of cycles, one could take the homology class
8217:
3617:
1400:
115:
17:
14132:
12848:
12148:
10150:
5970:
5875:
11568:
All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps
14765:
14418:
13598:
12191:
11793:
5215:
1147:
1105:
12757:
can be thought of as a gluing operation. Making a cut and then re-gluing it does not change the surface, so
11571:
11034:
10736:
10677:
7611:
1795:
1470:
5831:
1971:
14309:. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.
13818:
12210:
to perform homology-preserving reductions of the input cell complexes before resorting to matrix algebra.
10795:
3745:
1587:
1170:
distinguished by examining their "holes." For instance, a figure-eight shape has more holes than a circle
1127:. (Simplices are a generalization of triangles to arbitrary dimension; for example, an edge in a graph is
13638:
12203:
11666:
7691:
7028:
6866:
6820:
6629:
4936:
2102:
1262:
circles in a figure-eight shape provide examples of one-dimensional cycles, or 1-cycles, and the 2-torus
14773:
10243:
8810:
7820:
5440:
3966:
if and only if they differ by the addition of a boundary. This also implies that the "zero" element of
3298:
3266:
490:
189:
13501:. Chain complexes (since greatly generalized) form the basis for most modern treatments of homology.
11764:
5680:
4215:
14313:
Hilton, Peter (1988), "A Brief, Subjective
History of Homology and Homotopy Theory in This Century",
12067:
31:
14241:. Memoirs of the American Mathematical Society number. Vol. 55. American Mathematical Society.
13949:
5744:
5533:
5306:
5146:
4820:
4679:
4428:
3354:
are different from the boundaries of "filled in" holes, because the homology of a topological space
14906:
14572:
13618:
11185:
10608:
9477:
6308:
5633:
2316:
72:
to various other types of mathematical objects. Lastly, since there are many homology theories for
7798:
7654:
5872:
corresponds, as in the previous examples, to the fact that there is a single connected component.
4493:
4137:
14858:
14844:
14422:
13744:
13704:
12215:
11930:
11274:
and, especially in algebraic topology, this provides two ways to compute the important invariant
8873:
8776:
8704:
8446:
8320:
2519:
13921:
13491:
is regarded as the empty connected sum. Homology is preserved by the operation of connected sum.
11680:
10497:
9820:
2477:
2444:
14793:
14714:
14591:
14552:
14512:
13944:
13516:
12871:
12144:
12140:
12136:
10671:
7502:
7208:{\displaystyle H_{n}(X):=\ker(\partial _{n})/\mathrm {im} (\partial _{n+1})=Z_{n}(X)/B_{n}(X),}
1060:
92:, giving rise to various cohomology theories, in addition to the notion of the cohomology of a
14788:
9057:
8740:
7895:
7859:
7772:
7736:
6468:
4221:
4169:
3969:
3933:
3869:
3706:
1331:
49:, has three primary, closely-related usages. The most direct usage of the term is to take the
14635:
14562:
14489:
N.J. Windberger intro to algebraic topology, last six lectures with an easy intro to homology
14479:
14164:
13866:
13623:
12863:-gon. Variations are also possible, for example a hexagon may also be glued to form a torus.
12207:
11903:
11713:
10633:
10079:{\displaystyle \partial _{n}(c)=\sum _{\sigma _{i}\in X_{n}}m_{i}\partial _{n}(\sigma _{i}).}
9428:
9268:
6503:
The statement that the boundary of a boundary is trivial is equivalent to the statement that
6435:
11841:
10946:
10565:
8577:
8026:
5075:
4750:
2069:
14783:
14735:
14709:
14557:
14487:
13985:
13936:
13881:
13467:
13183:
13127:
13071:
13015:
12958:
12901:
12673:
12604:
12520:
12483:
12473:
12437:
12401:
12363:
12323:
12285:
12244:
11750:
11302:
11277:
11099:
10447:
10416:
10389:
10292:
10212:
10106:
9929:
8396:
5409:
4991:
4898:
4612:
4515:
4357:
4320:
4282:
4038:
3562:
3531:
3484:
3437:
3228:
3201:
3170:
2996:
2969:
2942:
2915:
2888:
2861:
2834:
2807:
2780:
2753:
2726:
2699:
2672:
2645:
2618:
2591:
2559:
2174:
2043:
1909:
1882:
1556:
1529:
1292:
1265:
1227:
1200:
1173:
1077:
983:
932:
903:
878:
729:
572:
460:
198:
164:
14293:
12032:
9787:
8:
14630:
14414:
13608:
13515:
further developed the theory of algebraic homology groups in the period 1925β28. The new
12252:
12083:
11934:
11355:
10653:
10623:
9231:
9220:
8819:
8603:
8440:, under the group operation of concatenation. The most fundamental homotopy group is the
7989:
7933:
6623:
5940:
5626:
3701:
2096:
1876:
1672:
1120:
1041:
1012:
977:
754:
81:
14834:
13989:
13940:
13885:
13644:
12656:
happens to cross over the twist given to one join. If an equidistant cut on one side of
1258:
that represent homology classes (the elements of homology groups). For example, the two
1224:(a 2-dimensional surface shaped like an inner tube) has different holes from a 2-sphere
14828:
14798:
14778:
14699:
14689:
14567:
14547:
14330:
14001:
13899:
13593:
13575:
13527:
12109:
11893:
9457:
9250:
9236:
9226:
9037:
9017:
8845:
8482:
8423:
8376:
8356:
7977:
7950:
6027:
4986:
4600:
4005:
3913:
3849:
3829:
3809:
3683:
3663:
3597:
3511:
3464:
3417:
3397:
3377:
3357:
3337:
3150:
3130:
3110:
3090:
3070:
3050:
3030:
2424:
2291:
1872:
1854:
1639:
1581:
1446:
1367:
1155:
959:
912:
907:
706:
549:
440:
193:
46:
13633:
12211:
2011:
1939:
1151:
1115:
For 1-dimensional topological spaces, probably the simplest homology theory to use is
14823:
14816:
14682:
14640:
14505:
14459:
14436:
14426:
14392:
14362:
14299:
14278:
14268:
14252:
14242:
14225:
14215:
14053:
14005:
13569:
12805:
12633:
12508:
12263:'s proof in 1871 of the independence of "homology numbers" from the choice of basis.
12172:
12116:
relates the dynamics of a gradient flow on a manifold to, for example, its homology.
12075:
11641:
10936:
from the category of chain complexes to the category of abelian groups (or modules).
10658:
10643:
10613:
10132:
10122:
10110:
9262:
8867:
8441:
7997:
4031:
3907:
1871:. In topology, the boundary of a space is technically obtained by taking the space's
1136:
1132:
1109:
1037:
1029:
227:
93:
89:
73:
14848:
14448:
13903:
12867:
14596:
14542:
14384:
14326:
14322:
14207:
13993:
13954:
13889:
13508:
13272:
13153:
12855:-gons) can be glued to make different manifolds. Conversely, a closed surface with
12668:
12248:
12152:
12101:
12094:
10186:
10180:
8500:
7274:
6666:
5671:
5209:
3334:
On the other hand, the boundaries discussed in the homology of a topological space
3261:
1081:
1069:
1045:
597:
13844:
14655:
14650:
14492:
14174:
14047:
12195:
12079:
11953:
11945:
11189:
10618:
10552:
10537:
10190:
8863:
8020:
7022:
5622:
4214:-dimensional boundary. A 0-dimensional-boundary hole is simply a gap between two
4203:
3024:
2008:, and with respect to suitable orientation conventions, the oriented boundary of
1065:
1052:
1008:
14838:
14388:
12661:
12214:
is written in Lisp, and in addition to homology it may also be used to generate
12202:
are also written in C++. All three implement pre-processing algorithms based on
14745:
14677:
14376:
14341:
14264:
14138:, in French, note 41, explicitly names Noether as inventing the homology group.
13628:
13547:
12125:
12117:
12090:
11835:
10648:
10628:
10556:
8809:. Higher homotopy groups are sometimes difficult to compute. For instance, the
8627:
8315:
7251:
7065:
5936:
4262:
3740:
1143:
1116:
1085:
1033:
543:
485:
13997:
13894:
7598:{\displaystyle \epsilon \left(\sum _{i}n_{i}\sigma _{i}\right)=\sum _{i}n_{i}}
1785:{\displaystyle B_{n}:=\mathrm {im} \,d_{n+1}:=\{d_{n+1}(c)\,|\;c\in C_{n+1}\}}
1142:
There are also other ways of computing these homology groups, for example via
868:{\displaystyle B_{n}:=\mathrm {im} \,d_{n+1}:=\{d_{n+1}(c)\,|\;c\in C_{n+1}\}}
14900:
14755:
14665:
14645:
14440:
14354:
13520:
13512:
13498:
13453:
11878:
11637:
8019:
Cohomology groups are formally similar to homology groups: one starts with a
5952:
4539:
4022:-dimensional boundaries, which also includes formal sums of such boundaries.
3195:
1664:
1464:
1394:
1128:
1123:, the latter of which involves a decomposition of the topological space into
1073:
159:
105:
57:
52:
13974:"Finite element exterior calculus, homological techniques, and applications"
13530:". Algebraic homology remains the primary method of classifying manifolds.
9975:
is defined over (usually integers, unless otherwise specified). Then define
4311:
14740:
14660:
14606:
14406:
14383:, Graduate Texts in Mathematics, vol. 72, Springer, pp. 169β184,
14256:
14214:. Princeton mathematical series. Vol. 19. Princeton University Press.
14203:
13919:
13583:
13504:
13209:
12912:
12883:
12738:
12629:
12466:
12260:
12243:
Homology theory can be said to start with the Euler polyhedron formula, or
12199:
12147:
may need to be solved on topologically nontrivial domains, for example, in
12129:
12113:
6967:{\displaystyle \mathrm {im} (\partial _{n+1})\subseteq \ker(\partial _{n})}
6568:{\displaystyle \mathrm {im} (\partial _{n+1})\subseteq \ker(\partial _{n})}
5946:
The general construction begins with an object such as a topological space
5707:
4927:
223:
14229:
9195:{\displaystyle H_{1}(X)=\{a_{l}l+a_{r}r\,|\;a_{l},a_{r}\in \mathbb {Z} \}}
8016:, and agree with the simplicial homology groups for a simplicial complex.
3252:.) However, it is sometimes desirable to restrict to nicer spaces such as
14750:
13588:
12889:
The complete classification of 1- and 2-manifolds is given in the table.
12698:
12226:
includes a homology solver for finite element meshes, which can generate
12121:
12071:
10194:
1048:, there are multiple common methods to compute the same homology groups.
38:
4273:
14694:
14625:
14584:
14334:
12808:. Similarly, in the projective plane, following the unshrinkable cycle
12227:
11896:: any continuous vector field on the 2-sphere (or more generally, the 2
10996:
10638:
10237:
10149:
to be the free abelian group (or free module) whose generators are all
9603:
8907:
8373:
is the group of homotopy classes of basepoint-preserving maps from the
85:
13958:
13865:
van den Berg, J.B.; Ghrist, R.; Vandervorst, R.C.; WΓ³jcik, W. (2015).
12108:, PoincarΓ© was one of the first to consider the interplay between the
14719:
13449:
For a non-orientable surface, a hole is equivalent to two cross-caps.
13313:
12697:
has both joins twisted. The uncut form, generally represented as the
11832:
11344:{\displaystyle 0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0}
8023:, which is the same as a chain complex but whose arrows, now denoted
3556:
3328:
3257:
3023:
More generally, any shape with no boundary can be "filled in" with a
1259:
1254:
Studying topological features such as these led to the notion of the
14177:
gives documentation (translated into
English from French originals).
12392:
1015:
can be used to relate homology groups of different chain complexes.
14704:
14672:
14621:
14528:
13972:
Arnold, Douglas N.; Richard S. Falk; Ragnar
Winther (16 May 2006).
13971:
13864:
12219:
11938:
11758:
3253:
1167:
1076:. One can describe this latter construction explicitly in terms of
12784:-cycle ends are glued together with opposite orientations. Hence 2
12780:. This is because the Klein bottle is made from a cylinder, whose
12737:
on the torus were when it was cut open and flattened down. In the
12465:
in the diagram can be shrunk to the pole, and even the equatorial
12233:
1161:
13739:
13699:
12511:, so as a consequence, it also has trivial first homology group.
12105:
10985:
10933:
10158:
9810:
This behavior on the generators induces a homomorphism on all of
1124:
1056:
13922:"Homology and Cohomology Computation in Finite Element Modeling"
12276:
2414:{\displaystyle Z_{n}:=\ker d_{n}:=\{c\in C_{n}\,|\;d_{n}(c)=0\}}
1100:
Perhaps the most familiar usage of the term homology is for the
696:{\displaystyle Z_{n}:=\ker d_{n}:=\{c\in C_{n}\,|\;d_{n}(c)=0\}}
13822:
13041:
12984:
12652:
forwards right round and back reverses left and right, because
12594:
12432:
12352:
12314:
10127:
Using simplicial homology example as a model, one can define a
6422:{\displaystyle \partial _{n}\circ \partial _{n+1}=0_{n+1,n-1},}
4607:
4382:
4352:
3614:, a choice of appropriate homology theory, and a chain complex
1316:
represent 2-cycles. Cycles form a group under the operation of
64:
This operation, in turn, allows one to associate various named
12128:
can follow complex trajectories; in particular, they may form
11661:
Notable theorems proved using homology include the following:
3394:, and not with new shapes built from gluing extra pieces onto
1323:
14346:
Euler's Gem: The
Polyhedron Formula and the Birth of Topology
13920:
Pellikka, M; S. Suuriniemi; L. Kettunen; C. Geuzaine (2013).
13097:
12515:
12188:
12168:
10992:
belongs to into the category of abelian groups (or modules).
9007:{\displaystyle H_{1}(X)\cong \mathbb {Z} \times \mathbb {Z} }
5967:. A chain complex is a sequence of abelian groups or modules
5205:
4603:
representing the one-dimensional hole contained in a circle.
1119:, which could be regarded as a 1-dimensional special case of
12176:
10444:
to this sequence, one obtains a chain complex; the homology
9919:{\textstyle c=\sum _{\sigma _{i}\in X_{n}}m_{i}\sigma _{i},}
7266:
therefore measure "how far" the chain complex associated to
3331:
classes and therefore cannot be "filled in" with manifolds.
3167:
would be the boundary of that cone. (For example, a cone on
14497:
13973:
12866:
The first recognisable theory of homology was published by
12223:
12184:
12180:
11264:{\displaystyle \chi =\sum (-1)^{n}\,\mathrm {rank} (H_{n})}
11174:{\displaystyle \chi =\sum (-1)^{n}\,\mathrm {rank} (A_{n})}
11023:
belongs to into the category of abelian groups or modules.
11007:, and that therefore the homology groups (which are called
8701:
is surjective and its kernel is the commutator subgroup of
7285:) are defined as homologies of the augmented chain complex
5813:
5605:
5382:
5188:
4868:
4727:
4476:
76:
that produce the same answer, one also often speaks of the
8955:{\displaystyle \pi _{1}(X)\cong \mathbb {Z} *\mathbb {Z} }
8192:{\displaystyle \mathrm {im} \left(d^{n-1}\right)=B^{n}(X)}
12893:
Topological characteristics of closed 1- and 2-manifolds
12171:
library for performing fast matrix operations, including
12061:
4889:
3866:-dimensional boundaries. In other words, the elements of
1467:, and are defined in terms of the boundary homomorphisms
14494:
Algebraic topology Allen
Hatcher - Chapter 2 on homology
14263:
13680:
12556:
can be deformed into one another. In particular, cycles
12151:. In these simulations, solution is aided by fixing the
9592:{\displaystyle \sigma =(\sigma ,\sigma ,\dots ,\sigma )}
9418:{\displaystyle \sigma <\sigma <\cdots <\sigma }
9261:. The orientation is captured by ordering the complex's
2939:. This phenomenon is sometimes described as saying that
2281:{\displaystyle d_{1}(k\cdot )=k\cdot \{1\}-k\cdot \{0\}}
1072:, measuring the failure of an appropriate functor to be
12583:, and the other opposite pair represents the cut along
10386:
Continuing in this fashion, a sequence of free modules
10379:{\displaystyle p_{2}:F_{2}\to \ker \left(p_{1}\right).}
10197:. Here one starts with some covariant additive functor
6731:{\displaystyle B_{n}(X)=\mathrm {im} (\partial _{n+1})}
13799:
12480:
can be similarly shrunk to a point. This implies that
12112:
of a dynamical system and its topological invariants.
10999:
is that in cohomology the chain complexes depend in a
9856:
6817:
is abelian all its subgroups are normal. Then because
2993:-shaped "hole" or that it could be "filled in" with a
13676:
13674:
13672:
13470:
13186:
13130:
13074:
13018:
12961:
12832:-cycle around twice in the Klein bottle gives simply
12812:
round twice remarkably creates a trivial cycle which
12676:
12607:
12523:
12486:
12440:
12404:
12366:
12326:
12288:
12259:-fold connectedness numerical invariants in 1857 and
12120:
extended this to infinite-dimensional manifolds. The
12035:
12000:
11965:
11906:
11844:
11796:
11767:
11716:
11683:
11574:
11367:
11313:
11280:
11201:
11111:
11090:) is a chain complex such that all but finitely many
11037:
10949:
10851:
10798:
10739:
10680:
10568:
10500:
10450:
10419:
10392:
10322:
10295:
10246:
10215:
9984:
9932:
9823:
9790:
9614:
9528:
9480:
9460:
9431:
9366:
9295:
9271:
9096:
9060:
9040:
9020:
8968:
8916:
8876:
8848:
8822:
8779:
8743:
8707:
8636:
8606:
8580:
8509:
8485:
8449:
8426:
8399:
8379:
8359:
8323:
8220:
8134:
8067:
8029:
7898:
7862:
7823:
7801:
7775:
7739:
7694:
7657:
7614:
7528:
7505:
7294:
7076:
7031:
6980:
6908:
6869:
6823:
6748:
6675:
6632:
6581:
6509:
6471:
6438:
6355:
6311:
6104:
6035:
5973:
5878:
5834:
5716:
5683:
5641:
5498:
5443:
5412:
5271:
5218:
5109:
5078:
5021:
4994:
4939:
4901:
4783:
4753:
4642:
4615:
4548:
4518:
4496:
4391:
4360:
4323:
4285:
4224:
4172:
4140:
4054:
4008:
3972:
3936:
3930:-cycles, and any two cycles are regarded as equal in
3916:
3872:
3852:
3832:
3812:
3748:
3709:
3686:
3666:
3620:
3600:
3565:
3534:
3514:
3487:
3467:
3440:
3420:
3400:
3380:
3360:
3340:
3301:
3269:
3231:
3204:
3173:
3153:
3133:
3113:
3093:
3073:
3053:
3033:
2999:
2972:
2945:
2918:
2891:
2864:
2837:
2810:
2783:
2756:
2729:
2702:
2675:
2648:
2621:
2594:
2562:
2522:
2480:
2447:
2427:
2324:
2294:
2204:
2177:
2105:
2072:
2046:
2014:
1974:
1942:
1912:
1885:
1857:
1798:
1680:
1642:
1590:
1559:
1532:
1473:
1449:
1403:
1370:
1334:
1295:
1268:
1230:
1203:
1176:
986:
962:
935:
915:
881:
763:
732:
709:
606:
575:
552:
493:
463:
443:
239:
201:
167:
118:
13565:
12164:
5061:{\displaystyle H_{0}\left(B^{2}\right)=\mathbb {Z} }
4588:{\displaystyle H_{1}\left(S^{1}\right)=\mathbb {Z} }
3256:, and not every cone is homeomorphic to a manifold.
1251:(a 2-dimensional surface shaped like a basketball).
14321:(5), Mathematical Association of America: 282β291,
11959:Invariance of dimension: if non-empty open subsets
10910:{\displaystyle f_{n-1}\circ d_{n}=e_{n}\circ f_{n}}
1936:Topologically, the boundary of the closed interval
226:) such that the composition of any two consecutive
27:
Applying
Algebraic structures to topological spaces
13716:
13714:
13669:
13483:
13199:
13143:
13087:
13031:
12974:
12689:
12620:
12536:
12499:
12453:
12417:
12379:
12339:
12301:
12230:bases directly usable by finite element software.
12050:
12021:
11986:
11918:
11865:
11823:
11782:
11740:
11702:
11656:
11624:
11557:
11343:
11286:
11263:
11173:
11082:
10961:
10909:
10837:
10784:
10725:
10591:
10528:
10463:
10432:
10405:
10378:
10308:
10281:
10228:
10078:
9945:
9918:
9842:
9802:
9773:
9591:
9493:
9466:
9446:
9417:
9352:
9277:
9194:
9082:
9046:
9026:
9006:
8954:
8898:
8854:
8834:
8801:
8765:
8729:
8693:
8618:
8592:
8566:
8491:
8471:
8432:
8412:
8385:
8365:
8345:
8291:
8191:
8116:
8045:
7920:
7884:
7841:
7809:
7787:
7761:
7725:
7673:
7643:
7597:
7511:
7488:
7207:
7056:
7013:
6966:
6894:
6848:
6795:
6730:
6657:
6614:
6567:
6493:
6457:
6421:
6330:
6294:
6083:
6018:
5930:
5915:
5864:
5819:
5698:
5660:
5611:
5482:
5425:
5388:
5257:
5194:
5094:
5060:
5007:
4971:
4914:
4874:
4769:
4733:
4628:
4587:
4530:
4504:
4482:
4373:
4336:
4298:
4246:
4194:
4158:
4127:{\displaystyle H_{0}(X),H_{1}(X),H_{2}(X),\ldots }
4126:
4014:
3994:
3958:
3922:
3894:
3858:
3838:
3818:
3798:
3731:
3692:
3680:that is compatible with that homology theory, the
3672:
3652:
3606:
3578:
3547:
3528:gives rise to a boundary class in the homology of
3520:
3500:
3473:
3453:
3426:
3406:
3386:
3366:
3346:
3319:
3287:
3244:
3217:
3186:
3159:
3139:
3119:
3099:
3079:
3059:
3047:has no boundary, then the boundary of the cone on
3039:
3012:
2985:
2958:
2931:
2904:
2877:
2850:
2823:
2796:
2769:
2742:
2715:
2688:
2661:
2634:
2607:
2575:
2548:
2508:
2466:
2433:
2413:
2300:
2280:
2190:
2163:
2087:
2058:
2032:
2000:
1960:
1925:
1898:
1863:
1843:
1784:
1648:
1628:
1572:
1545:
1518:
1455:
1435:
1376:
1356:
1308:
1281:
1243:
1216:
1189:
1095:
999:
968:
948:
921:
894:
867:
745:
715:
695:
588:
558:
534:
476:
449:
426:
214:
180:
150:
14381:Classical Topology and Combinatorial Group Theory
10943:in a covariant manner (meaning that any morphism
9360:of its vertices listed in increasing order (i.e.
8600:, this homomorphism can be complicated, but when
14898:
10209:is defined as follows: first find a free module
9353:{\displaystyle (\sigma ,\sigma ,\dots ,\sigma )}
8117:{\displaystyle \ker \left(d^{n}\right)=Z^{n}(X)}
6432:i.e., the constant map sending every element of
5661:{\displaystyle \mathbb {Z} \times \mathbb {Z} .}
14202:
13915:
13913:
13711:
12234:Some non-homology-based discussions of surfaces
12132:that can be investigated using Floer homology.
9474:th vertex appearing in the tuple). The mapping
7957:are defined using the simplicial chain complex
7817:in the chain complex represents the unique map
7258:+1)th map is always equal to the kernel of the
6084:{\displaystyle \partial _{n}:C_{n}\to C_{n-1},}
4344:is the outer shell, not the interior, of a ball
2040:is given by the union of a positively-oriented
1162:Inspirations for homology (informal discussion)
14413:, Israel Mathematical Conference Proceedings,
14379:(1993), "Homology Theory and Abelianization",
14236:
14134:L'Γ©mergence de la notion de groupe d'homologie
13858:
13614:Homological conjectures in commutative algebra
12097:to evaluate, for instance, holes in coverage.
10969:induces a morphism from the chain complex of
10109:representations of these boundary mappings in
7014:{\displaystyle \mathrm {im} (\partial _{n+1})}
6615:{\displaystyle \mathrm {im} (\partial _{n+1})}
99:
14513:
13452:Any closed 2-manifold can be realised as the
9034:of the lefthand cycle and the homology class
8694:{\displaystyle h_{*}:\pi _{1}(X)\to H_{1}(X)}
8567:{\displaystyle h_{*}:\pi _{n}(X)\to H_{n}(X)}
5558:
5545:
1080:, or more abstractly from the perspective of
13910:
13735:"Delta complexes, Betti numbers and torsion"
9189:
9119:
9054:of the righthand cycle as basis elements of
6796:{\displaystyle Z_{n}(X)=\ker(\partial _{n})}
5800:
5794:
5592:
5586:
5369:
5363:
5175:
5169:
4855:
4849:
4714:
4708:
4525:
4519:
4463:
4457:
3594:Given a sufficiently-nice topological space
2408:
2357:
2275:
2269:
2257:
2251:
2158:
2152:
2146:
2140:
2079:
2073:
2053:
2047:
1995:
1989:
1981:
1975:
1779:
1722:
1150:when applied to a cohomology theory such as
862:
805:
690:
639:
14239:Foundations of relative homological algebra
14089:
14066:
12729:Cycles can be joined or added together, as
12398:Cycles on a hemispherical projective plane
12022:{\displaystyle V\subseteq \mathbb {R} ^{n}}
11987:{\displaystyle U\subseteq \mathbb {R} ^{m}}
10939:If the chain complex depends on the object
10171:arise from the boundary maps of simplices.
8626:, the Hurewicz homomorphism coincides with
8292:{\displaystyle H^{n}(X)=Z^{n}(X)/B^{n}(X),}
4254:describes the path-connected components of
3653:{\displaystyle (C_{\bullet },d_{\bullet })}
1436:{\displaystyle (C_{\bullet },d_{\bullet })}
1324:Cycles and Boundaries (informal discussion)
151:{\displaystyle (C_{\bullet },d_{\bullet })}
14881:
14854:
14520:
14506:
13965:
13732:
13692:
12564:cannot be shrunk to a point whereas cycle
10601:
9157:
2382:
1756:
839:
664:
14375:
14018:
13948:
13893:
12709:are identified as the same point. Again,
12143:for differential equations involving the
12009:
11974:
11811:
11770:
11230:
11140:
10995:The only difference between homology and
10097:turns out to be the number of "holes" in
9211:overall homology functor for the theory.
9185:
9151:
9000:
8992:
8948:
8940:
8866:. As usual, its first homotopy group, or
7803:
7481:
7473:
7465:
7435:
7405:
7376:
7340:
7304:
6275:
6245:
6215:
6186:
6150:
6114:
6019:{\displaystyle C_{0},C_{1},C_{2},\ldots }
5916:{\displaystyle H_{1}(P)=\mathbb {Z} _{2}}
5903:
5858:
5769:
5748:
5686:
5651:
5643:
5538:
5344:
5336:
5310:
5150:
5054:
4824:
4683:
4581:
4498:
4432:
3307:
3304:
3275:
3272:
2376:
1988:
1984:
1750:
1702:
833:
785:
658:
84:.) There is also a related notion of the
14340:
14186:
14095:
14084:
14078:
14072:
14041:
14039:
13519:formally treated topological classes as
12648:can be. But unlike the torus, following
12593:
10440:can be defined. By applying the functor
9425:in the complex's vertex ordering, where
8309:
8012:) are defined for any topological space
4261:For the homology groups of a graph, see
14405:
14353:
14291:
14237:Eilenberg, Samuel; Moore, J.C. (1965).
14152:
13811:
13805:
13793:
13781:
13769:
13757:
13720:
13663:
12571:If the torus surface is cut along both
12222:groups of finite simplicial complexes.
11824:{\displaystyle f:U\to \mathbb {R} ^{n}}
11677:to itself, then there is a fixed point
8773:is isomorphic to the abelianization of
5258:{\displaystyle T^{2}=S^{1}\times S^{1}}
1108:yields the same homology groups as the
14:
14899:
14446:
14312:
14274:The Princeton Companion to Mathematics
14119:
14107:
14030:
13681:Gowers, Barrow-Green & Leader 2010
13641:- also has a list of homology theories
13464:projective planes, where the 2-sphere
12062:Application in science and engineering
11625:{\displaystyle H_{n}(C)\to H_{n-1}(A)}
11298:which gave rise to the chain complex.
11188:in the case of abelian groups and the
11083:{\displaystyle d_{n}:A_{n}\to A_{n-1}}
10785:{\displaystyle e_{n}:B_{n}\to B_{n-1}}
10726:{\displaystyle d_{n}:A_{n}\to A_{n-1}}
9214:
8574:called the Hurewicz homomorphism. For
8203:follow from the same description. The
7644:{\displaystyle \sum n_{i}\sigma _{i},}
6305:where 0 denotes the trivial group and
2696:, one could glue a 2-dimensional disk
1844:{\displaystyle d_{n}:C_{n}\to C_{n-1}}
1519:{\displaystyle d_{n}:C_{n}\to C_{n-1}}
1154:or (in the case of real coefficients)
14501:
14045:
14036:
10674:: A morphism from the chain complex (
8053:point in the direction of increasing
5865:{\displaystyle H_{0}(P)=\mathbb {Z} }
4202:describes, informally, the number of
2311:In the context of chain complexes, a
2001:{\displaystyle \{0\}\,\amalg \,\{1\}}
14449:"28. History of Homological Algebra"
12859:non-zero classes can be cut into a 2
12476:shows that any closed curve such as
11673:is any continuous map from the ball
10838:{\displaystyle f_{n}:A_{n}\to B_{n}}
10665:
10116:
9850:, write it as the sum of generators
9205:
5677:, a simple computation shows (where
4025:
3799:{\displaystyle H_{n}(X)=Z_{n}/B_{n}}
1629:{\displaystyle d_{n-1}\circ d_{n}=0}
1018:
12776:-cycle), and it will come back as β
12472:can be shrunk in the same way. The
11354:of chain complexes gives rise to a
10494:A common use of group (co)homology
9265:and expressing an oriented simplex
9257:-dimensional oriented simplexes of
9253:or module whose generators are the
7726:{\displaystyle {\tilde {H}}_{i}(X)}
7057:{\displaystyle \ker(\partial _{n})}
6895:{\displaystyle \ker(\partial _{n})}
6849:{\displaystyle \ker(\partial _{n})}
6658:{\displaystyle \ker(\partial _{n})}
4972:{\displaystyle T=S^{1}\times S^{1}}
3414:. For example, any embedded circle
2164:{\displaystyle d_{1}()=\{1\}-\{0\}}
24:
14196:
11241:
11238:
11235:
11232:
11151:
11148:
11145:
11142:
10048:
9986:
9616:
9482:
9225:The motivating example comes from
8139:
8136:
7827:
7681:which are the fixed generators of
7438:
7408:
7379:
7343:
7307:
7138:
7130:
7127:
7109:
7042:
6993:
6985:
6982:
6952:
6921:
6913:
6910:
6880:
6834:
6781:
6710:
6702:
6699:
6643:
6594:
6586:
6583:
6553:
6522:
6514:
6511:
6370:
6357:
6278:
6248:
6218:
6189:
6153:
6117:
6037:
5549:
4150:
4147:
3589:
3461:already bounds some embedded disk
3374:has to do with the original space
1698:
1695:
781:
778:
484:of this chain complex is then the
25:
14918:
14473:
13874:Journal of Differential Equations
13845:"Robert Ghrist: applied topology"
12804:) = 0. This phenomenon is called
10792:) is a sequence of homomorphisms
10282:{\displaystyle p_{1}:F_{1}\to X.}
10174:
7842:{\displaystyle \longrightarrow X}
5483:{\displaystyle T^{n}=(S^{1})^{n}}
3320:{\displaystyle \mathbb {CP} ^{2}}
3288:{\displaystyle \mathbb {RP} ^{2}}
1526:of the chain complex, where each
1146:, or by taking the output of the
1055:, a homology theory is a type of
535:{\displaystyle H_{n}=Z_{n}/B_{n}}
14880:
14853:
14843:
14833:
14822:
14812:
14811:
14605:
13747:from the original on 2021-12-11.
13707:from the original on 2021-12-11.
13568:
13007:
13004:
12749:goes round the opposite way. If
12391:
12351:
12313:
12275:
11783:{\displaystyle \mathbb {R} ^{n}}
11019:functors from the category that
10471:of this complex depends only on
10105:. It may be computed by putting
5699:{\displaystyle \mathbb {Z} _{2}}
5402:products of a topological space
4926:
4888:
4597:finitely-generated abelian group
4310:
4272:
2804:. Similarly, given a two-sphere
2198:is a homomorphism, this implies
2099:analog of this statement is that
14483:at Encyclopaedia of Mathematics
14180:
14166:Bourbaki and Algebraic Topology
14158:
14141:
14125:
14113:
14101:
14024:
14012:
13837:
13787:
12816:be shrunk to a point; that is,
12753:is thought of as a cut, then β
11657:Application in pure mathematics
11651:
11011:in this context and denoted by
9968:are coefficients from the ring
7262:th map. The homology groups of
5950:, on which one first defines a
5931:Construction of homology groups
1968:is given by the disjoint union
1663:For example, in the context of
1102:homology of a topological space
1096:Homology of a Topological Space
980:, resulting in homology groups
929:, and taking the boundary maps
388:
78:homology of a topological space
14327:10.1080/0025570X.1988.11977391
14298:, Cambridge University Press,
14277:, Princeton University Press,
13775:
13763:
13751:
13733:Wildberger, Norman J. (2012).
13726:
13693:Wildberger, Norman J. (2012).
13686:
13657:
13540:
12849:contractible topological space
12745:goes round one way and −
12070:, data sets are regarded as a
11860:
11854:
11806:
11726:
11720:
11619:
11613:
11594:
11591:
11585:
11549:
11546:
11540:
11521:
11518:
11512:
11493:
11490:
11484:
11465:
11462:
11456:
11437:
11434:
11428:
11415:
11412:
11406:
11393:
11390:
11384:
11371:
11335:
11329:
11323:
11317:
11258:
11245:
11221:
11211:
11168:
11155:
11131:
11121:
11061:
10953:
10822:
10763:
10704:
10523:
10511:
10346:
10316:and a surjective homomorphism
10270:
10189:, one uses homology to define
10070:
10057:
10001:
9995:
9760:
9754:
9739:
9727:
9718:
9706:
9691:
9685:
9668:
9658:
9631:
9625:
9586:
9583:
9577:
9562:
9556:
9547:
9541:
9535:
9441:
9435:
9412:
9406:
9391:
9385:
9376:
9370:
9347:
9344:
9338:
9323:
9317:
9308:
9302:
9296:
9153:
9113:
9107:
9077:
9071:
8985:
8979:
8933:
8927:
8893:
8887:
8796:
8790:
8760:
8754:
8724:
8718:
8688:
8682:
8669:
8666:
8660:
8561:
8555:
8542:
8539:
8533:
8466:
8460:
8340:
8334:
8283:
8277:
8259:
8253:
8237:
8231:
8186:
8180:
8111:
8105:
7915:
7909:
7879:
7873:
7833:
7830:
7824:
7756:
7750:
7720:
7714:
7702:
7688:. The reduced homology groups
7478:
7462:
7432:
7402:
7373:
7337:
7301:
7250:A chain complex is said to be
7199:
7193:
7175:
7169:
7153:
7134:
7118:
7105:
7093:
7087:
7051:
7038:
7008:
6989:
6961:
6948:
6936:
6917:
6889:
6876:
6843:
6830:
6790:
6777:
6765:
6759:
6725:
6706:
6692:
6686:
6652:
6639:
6609:
6590:
6562:
6549:
6537:
6518:
6272:
6242:
6212:
6183:
6147:
6111:
6059:
5895:
5889:
5851:
5845:
5733:
5727:
5522:
5509:
5471:
5457:
5295:
5282:
4241:
4235:
4189:
4183:
4115:
4109:
4093:
4087:
4071:
4065:
3989:
3983:
3953:
3947:
3889:
3883:
3765:
3759:
3726:
3720:
3647:
3621:
2497:
2491:
2399:
2393:
2378:
2239:
2236:
2224:
2215:
2134:
2131:
2119:
2116:
2027:
2015:
1955:
1943:
1822:
1752:
1747:
1741:
1497:
1430:
1404:
1351:
1345:
835:
830:
824:
681:
675:
660:
357:
318:
279:
256:
145:
119:
13:
1:
14419:American Mathematical Society
14271:; Leader, Imre, eds. (2010),
13650:
13604:Extraordinary homology theory
11026:
10932:can be viewed as a covariant
10289:Then one finds a free module
9817:as follows. Given an element
9494:{\displaystyle \partial _{n}}
6626:of the boundary operator and
6331:{\displaystyle C_{i}\equiv 0}
5963:) encoding information about
4512:is the group of integers and
3295:and complex projective plane
1792:of the boundary homomorphism
1553:is an abelian group, and the
1148:Universal Coefficient Theorem
56:, resulting in a sequence of
14527:
14411:The Heritage of Emmy Noether
13695:"More homology computations"
10536:is to classify the possible
8737:, with the consequence that
7810:{\displaystyle \mathbb {Z} }
7674:{\displaystyle \sigma _{i},}
7499:where the boundary operator
4505:{\displaystyle \mathbb {Z} }
4159:{\displaystyle k^{\rm {th}}}
3846:-dimensional cycles) modulo
3087:, and so if one "filled in"
7:
14447:Weibel, Charles A. (1999),
14389:10.1007/978-1-4612-4372-4_6
13639:List of cohomology theories
13561:
12266:
12204:simple-homotopy equivalence
12158:
12149:electromagnetic simulations
11667:Brouwer fixed point theorem
10479:and is, by definition, the
8899:{\displaystyle \pi _{1}(X)}
8802:{\displaystyle \pi _{1}(X)}
8730:{\displaystyle \pi _{1}(X)}
8472:{\displaystyle \pi _{1}(X)}
8346:{\displaystyle \pi _{n}(X)}
8211:is then the quotient group
2549:{\displaystyle S^{1},S^{2}}
2066:with a negatively oriented
1384:is defined in terms of the
188:(whose elements are called
100:Homology of Chain Complexes
45:, originally introduced in
10:
14923:
14774:Banach fixed-point theorem
14046:Weeks, Jeffrey R. (2001).
12636:). Like the torus, cycles
12238:
12175:; it interfaces with both
12074:sampling of a manifold or
11703:{\displaystyle a\in B^{n}}
10529:{\displaystyle H^{2}(G,M)}
10178:
10120:
9843:{\displaystyle c\in C_{n}}
9218:
8811:homotopy groups of spheres
7849:from the empty simplex to
7064:. Then one can create the
5627:Betti number#More examples
5433:, then in general, for an
4260:
3910:whose representatives are
2509:{\displaystyle d_{n}(c)=0}
2474:is a cycle if and only if
2467:{\displaystyle c\in C_{n}}
104:To take the homology of a
29:
14807:
14764:
14728:
14614:
14603:
14535:
14454:, in James, I. M. (ed.),
14361:, Springer, p. 155,
14149:Emmy Noether and Topology
13998:10.1017/S0962492906210018
13895:10.1016/j.jde.2015.03.022
13599:EilenbergβSteenrod axioms
12916:
12911:
12908:
12900:
12897:
12068:topological data analysis
11926:) vanishes at some point.
9784:which is considered 0 if
8503:describes a homomorphism
7512:{\displaystyle \epsilon }
6465:to the group identity in
5623:Torus#n-dimensional torus
4895:The solid disc or 2-ball
4002:is given by the group of
3027:, since if a given space
1106:Eilenberg-Steenrod axioms
32:Homology (disambiguation)
13619:Homological connectivity
13533:
12713:is non-shrinkable while
11646:Mayer-Vietoris sequences
11636:and are provided by the
11634:connecting homomorphisms
10973:to the chain complex of
10733:) to the chain complex (
10205:. The chain complex for
9083:{\displaystyle H_{1}(X)}
8766:{\displaystyle H_{1}(X)}
7921:{\displaystyle B_{n}(X)}
7885:{\displaystyle Z_{n}(X)}
7788:{\displaystyle i\neq 0.}
7762:{\displaystyle H_{i}(X)}
6494:{\displaystyle C_{n-1}.}
4777:the homology groups are
4247:{\displaystyle H_{0}(X)}
4195:{\displaystyle H_{k}(X)}
3995:{\displaystyle H_{n}(X)}
3959:{\displaystyle H_{n}(X)}
3895:{\displaystyle H_{n}(X)}
3732:{\displaystyle H_{n}(X)}
2912:is the boundary of that
2777:is the boundary of that
2097:simplicial chain complex
1357:{\displaystyle H_{n}(X)}
14423:Oxford University Press
14147:Hirzebruch, Friedrich,
14136:, Nicolas Basbois (PDF)
14033:, pp. 2β3 (in PDF)
12644:cannot be shrunk while
12247:. This was followed by
12141:boundary-value problems
12029:are homeomorphic, then
11919:{\displaystyle k\geq 1}
11741:{\displaystyle f(a)=a.}
10988:from the category that
10670:Chain complexes form a
10602:Other homology theories
10483:-th derived functor of
9447:{\displaystyle \sigma }
9278:{\displaystyle \sigma }
9242:. Here the chain group
9090:, allowing us to write
8353:of a topological space
8207:th cohomology group of
8057:rather than decreasing
7275:reduced homology groups
6810:Since each chain group
6458:{\displaystyle C_{n+1}}
4279:The circle or 1-sphere
3225:whose boundary is that
1364:of a topological space
14829:Mathematics portal
14729:Metrics and properties
14715:Second-countable space
14348:, Princeton University
14168:by John McCleary (PDF)
13867:"Braid Floer homology"
13517:combinatorial topology
13485:
13201:
13145:
13089:
13033:
12976:
12870:in his seminal paper "
12691:
12622:
12599:
12538:
12501:
12455:
12419:
12381:
12341:
12303:
12145:Hodge-Laplace operator
12137:finite element methods
12052:
12023:
11988:
11920:
11867:
11866:{\displaystyle V=f(U)}
11825:
11784:
11742:
11704:
11632:The latter are called
11626:
11559:
11345:
11288:
11265:
11175:
11084:
10963:
10962:{\displaystyle X\to Y}
10911:
10839:
10786:
10727:
10593:
10592:{\displaystyle G=E/M.}
10543:which contain a given
10530:
10465:
10434:
10407:
10380:
10310:
10283:
10230:
10138:. A chain complex for
10080:
9947:
9920:
9844:
9804:
9775:
9657:
9593:
9519:and sends the simplex
9495:
9468:
9448:
9419:
9354:
9279:
9196:
9084:
9048:
9028:
9008:
8956:
8900:
8856:
8836:
8803:
8767:
8731:
8695:
8620:
8594:
8593:{\displaystyle n>1}
8568:
8493:
8473:
8434:
8414:
8387:
8367:
8347:
8293:
8193:
8118:
8047:
8046:{\displaystyle d_{n},}
7922:
7886:
7843:
7811:
7789:
7763:
7727:
7675:
7645:
7599:
7513:
7490:
7209:
7058:
7015:
6968:
6902:is abelian, and since
6896:
6850:
6797:
6732:
6659:
6616:
6569:
6495:
6459:
6423:
6332:
6296:
6085:
6020:
5917:
5866:
5821:
5700:
5662:
5613:
5484:
5427:
5390:
5259:
5196:
5096:
5095:{\displaystyle B^{n},}
5062:
5009:
4973:
4916:
4876:
4771:
4770:{\displaystyle S^{n},}
4735:
4630:
4589:
4532:
4506:
4484:
4375:
4338:
4300:
4248:
4196:
4160:
4128:
4039:topological invariants
4016:
3996:
3960:
3924:
3896:
3860:
3840:
3820:
3800:
3733:
3694:
3674:
3654:
3608:
3580:
3549:
3522:
3502:
3475:
3455:
3428:
3408:
3388:
3368:
3348:
3321:
3289:
3246:
3219:
3188:
3161:
3141:
3121:
3107:by gluing the cone on
3101:
3081:
3061:
3041:
3014:
2987:
2960:
2933:
2906:
2879:
2852:
2831:, one can glue a ball
2825:
2798:
2771:
2744:
2717:
2690:
2663:
2636:
2609:
2577:
2550:
2510:
2468:
2435:
2415:
2315:is any element of the
2302:
2282:
2192:
2165:
2089:
2088:{\displaystyle \{0\}.}
2060:
2034:
2002:
1962:
1927:
1900:
1865:
1845:
1786:
1671:is any element of the
1650:
1630:
1574:
1547:
1520:
1457:
1437:
1378:
1358:
1310:
1283:
1245:
1218:
1191:
1001:
970:
950:
923:
896:
869:
747:
717:
697:
590:
560:
546:boundaries, where the
536:
478:
451:
428:
216:
182:
152:
13624:Homological dimension
13486:
13484:{\displaystyle S^{2}}
13202:
13200:{\displaystyle K^{2}}
13146:
13144:{\displaystyle P^{2}}
13090:
13088:{\displaystyle T^{2}}
13034:
13032:{\displaystyle S^{2}}
12977:
12975:{\displaystyle S^{1}}
12768:But now consider two
12692:
12690:{\displaystyle P^{2}}
12623:
12621:{\displaystyle K^{2}}
12597:
12539:
12537:{\displaystyle T^{2}}
12502:
12500:{\displaystyle S^{2}}
12456:
12454:{\displaystyle S^{2}}
12420:
12418:{\displaystyle P^{2}}
12382:
12380:{\displaystyle K^{2}}
12342:
12340:{\displaystyle T^{2}}
12304:
12302:{\displaystyle S^{2}}
12282:Cycles on a 2-sphere
12208:discrete Morse theory
12053:
12024:
11989:
11921:
11868:
11826:
11785:
11743:
11705:
11627:
11560:
11346:
11289:
11287:{\displaystyle \chi }
11266:
11176:
11085:
10964:
10912:
10840:
10787:
10728:
10634:Intersection homology
10594:
10531:
10466:
10464:{\displaystyle H_{n}}
10435:
10433:{\displaystyle p_{n}}
10408:
10406:{\displaystyle F_{n}}
10381:
10311:
10309:{\displaystyle F_{2}}
10284:
10231:
10229:{\displaystyle F_{1}}
10165:. The homomorphisms β
10142:is defined by taking
10089:The dimension of the
10081:
9948:
9946:{\displaystyle X_{n}}
9921:
9845:
9805:
9776:
9637:
9594:
9496:
9469:
9449:
9420:
9355:
9280:
9197:
9085:
9049:
9029:
9009:
8957:
8901:
8857:
8837:
8804:
8768:
8732:
8696:
8621:
8595:
8569:
8494:
8474:
8435:
8415:
8413:{\displaystyle S^{n}}
8388:
8368:
8348:
8310:Homology vs. homotopy
8294:
8194:
8119:
8048:
7923:
7887:
7844:
7812:
7790:
7764:
7728:
7676:
7646:
7600:
7514:
7491:
7270:is from being exact.
7254:if the image of the (
7223:th homology group of
7210:
7059:
7016:
6969:
6897:
6851:
6798:
6733:
6660:
6617:
6570:
6496:
6460:
6424:
6333:
6297:
6086:
6021:
5918:
5867:
5822:
5701:
5663:
5614:
5485:
5428:
5426:{\displaystyle X^{n}}
5391:
5260:
5197:
5097:
5068:. In general, for an
5063:
5010:
5008:{\displaystyle B^{2}}
4974:
4917:
4915:{\displaystyle B^{2}}
4877:
4772:
4736:
4631:
4629:{\displaystyle S^{2}}
4590:
4533:
4531:{\displaystyle \{0\}}
4507:
4485:
4376:
4374:{\displaystyle S^{1}}
4339:
4337:{\displaystyle S^{2}}
4301:
4299:{\displaystyle S^{1}}
4249:
4197:
4161:
4129:
4017:
3997:
3961:
3925:
3897:
3861:
3841:
3821:
3801:
3739:is then given by the
3734:
3695:
3675:
3655:
3609:
3581:
3579:{\displaystyle S^{1}}
3550:
3548:{\displaystyle S^{2}}
3523:
3503:
3501:{\displaystyle S^{2}}
3476:
3456:
3454:{\displaystyle S^{2}}
3429:
3409:
3389:
3369:
3349:
3322:
3290:
3247:
3245:{\displaystyle S^{1}}
3220:
3218:{\displaystyle D^{2}}
3189:
3187:{\displaystyle S^{1}}
3162:
3142:
3122:
3102:
3082:
3062:
3042:
3015:
3013:{\displaystyle B^{3}}
2988:
2986:{\displaystyle B^{3}}
2961:
2959:{\displaystyle S^{2}}
2934:
2932:{\displaystyle B^{3}}
2907:
2905:{\displaystyle S^{2}}
2880:
2878:{\displaystyle S^{2}}
2853:
2851:{\displaystyle B^{3}}
2826:
2824:{\displaystyle S^{2}}
2799:
2797:{\displaystyle D^{2}}
2772:
2770:{\displaystyle S^{1}}
2745:
2743:{\displaystyle S^{1}}
2718:
2716:{\displaystyle D^{2}}
2691:
2689:{\displaystyle S^{1}}
2664:
2662:{\displaystyle T^{2}}
2637:
2635:{\displaystyle S^{2}}
2610:
2608:{\displaystyle S^{1}}
2578:
2576:{\displaystyle T^{2}}
2551:
2511:
2469:
2436:
2416:
2303:
2283:
2193:
2191:{\displaystyle d_{1}}
2166:
2090:
2061:
2059:{\displaystyle \{1\}}
2035:
2003:
1963:
1928:
1926:{\displaystyle S^{1}}
1901:
1899:{\displaystyle D^{2}}
1866:
1846:
1787:
1651:
1631:
1575:
1573:{\displaystyle d_{n}}
1548:
1546:{\displaystyle C_{n}}
1521:
1458:
1438:
1379:
1359:
1311:
1309:{\displaystyle S^{2}}
1284:
1282:{\displaystyle T^{2}}
1246:
1244:{\displaystyle S^{2}}
1219:
1217:{\displaystyle T^{2}}
1192:
1190:{\displaystyle S^{1}}
1002:
1000:{\displaystyle H_{n}}
971:
951:
949:{\displaystyle d_{n}}
924:
897:
895:{\displaystyle C_{n}}
870:
748:
746:{\displaystyle B_{n}}
718:
698:
591:
589:{\displaystyle Z_{n}}
561:
537:
479:
477:{\displaystyle H_{n}}
452:
429:
217:
215:{\displaystyle d_{n}}
183:
181:{\displaystyle C_{n}}
153:
14784:Invariance of domain
14736:Euler characteristic
14710:Bundle (mathematics)
14315:Mathematics Magazine
14292:Hatcher, A. (2002),
13507:and, independently,
13468:
13184:
13128:
13072:
13016:
12959:
12917:Torsion coefficient
12674:
12605:
12521:
12484:
12474:Jordan curve theorem
12438:
12402:
12364:
12324:
12286:
12245:Euler characteristic
12051:{\displaystyle m=n.}
12033:
11998:
11963:
11904:
11842:
11794:
11765:
11751:Invariance of domain
11714:
11681:
11572:
11365:
11311:
11303:short exact sequence
11278:
11199:
11109:
11100:Euler characteristic
11035:
10947:
10849:
10796:
10737:
10678:
10609:BorelβMoore homology
10566:
10498:
10448:
10417:
10390:
10320:
10293:
10244:
10213:
9982:
9930:
9854:
9821:
9803:{\displaystyle n=0.}
9788:
9612:
9526:
9478:
9458:
9429:
9364:
9293:
9269:
9094:
9058:
9038:
9018:
8966:
8914:
8874:
8846:
8820:
8777:
8741:
8705:
8634:
8604:
8578:
8507:
8483:
8447:
8424:
8397:
8377:
8357:
8321:
8302:in analogy with the
8218:
8132:
8065:
8027:
7896:
7860:
7856:Computing the cycle
7821:
7799:
7773:
7737:
7692:
7655:
7612:
7526:
7503:
7292:
7074:
7029:
6978:
6906:
6867:
6821:
6746:
6673:
6630:
6579:
6507:
6469:
6436:
6353:
6309:
6102:
6033:
5971:
5876:
5832:
5714:
5681:
5639:
5496:
5441:
5410:
5269:
5216:
5107:
5076:
5019:
4992:
4937:
4899:
4781:
4751:
4747:-dimensional sphere
4640:
4613:
4546:
4516:
4494:
4389:
4358:
4321:
4283:
4222:
4170:
4138:
4052:
4006:
3970:
3934:
3914:
3870:
3850:
3830:
3810:
3746:
3707:
3684:
3664:
3618:
3598:
3563:
3532:
3512:
3485:
3465:
3438:
3418:
3398:
3378:
3358:
3338:
3299:
3267:
3229:
3202:
3171:
3151:
3131:
3111:
3091:
3071:
3051:
3031:
2997:
2970:
2943:
2916:
2889:
2862:
2835:
2808:
2781:
2754:
2727:
2700:
2673:
2646:
2619:
2592:
2560:
2520:
2478:
2445:
2425:
2322:
2292:
2202:
2175:
2103:
2070:
2044:
2012:
1972:
1940:
1910:
1883:
1855:
1796:
1678:
1640:
1588:
1557:
1530:
1471:
1447:
1401:
1368:
1332:
1293:
1266:
1228:
1201:
1174:
984:
978:module homomorphisms
960:
933:
913:
879:
761:
730:
707:
604:
573:
550:
491:
461:
441:
237:
199:
165:
116:
112:which is a sequence
108:, one starts with a
30:For other uses, see
14794:Tychonoff's theorem
14789:PoincarΓ© conjecture
14543:General (point-set)
14456:History of Topology
14415:Bar-Ilan University
14212:Homological Algebra
13990:2006AcNum..15....1A
13941:2013SJSC...35B1195P
13929:SIAM J. Sci. Comput
13886:2015JDE...259.1663V
13609:Homological algebra
12894:
12084:persistent homology
11935:continuous function
11931:BorsukβUlam theorem
11358:of homology groups
11356:long exact sequence
10654:Persistent homology
10624:Hochschild homology
9232:simplicial homology
9221:Simplicial homology
9215:Simplicial homology
8835:{\displaystyle n=1}
8619:{\displaystyle n=1}
8306:th homology group.
7990:simplicial homology
7934:simplicial homology
7277:of a chain complex
5941:simplicial homology
5629:for more details).
5437:-dimensional torus
4045:represented by its
3908:equivalence classes
1582:group homomorphisms
1121:simplicial homology
1051:In the language of
1042:Hochschild homology
1013:homological algebra
194:group homomorphisms
82:popular mathematics
14779:De Rham cohomology
14700:Polyhedral complex
14690:Simplicial complex
14359:Algebraic Topology
14295:Algebraic Topology
14269:Barrow-Green, June
14204:Cartan, Henri Paul
14173:2008-07-23 at the
14049:The Shape of Space
13935:(5): B1195βB1214.
13819:"CompTop overview"
13760:, pp. 105β106
13683:, pp. 390β391
13594:De Rham cohomology
13576:Mathematics portal
13528:algebraic topology
13481:
13197:
13141:
13085:
13029:
12972:
12892:
12876:J. Ecole polytech.
12687:
12618:
12600:
12534:
12497:
12451:
12415:
12377:
12337:
12320:Cycles on a torus
12299:
12194:2013-07-15 at the
12110:invariant manifold
12048:
12019:
11984:
11952:maps some pair of
11916:
11894:Hairy ball theorem
11863:
11821:
11780:
11738:
11700:
11622:
11555:
11341:
11284:
11261:
11171:
11080:
10959:
10907:
10835:
10782:
10723:
10589:
10526:
10461:
10430:
10413:and homomorphisms
10403:
10376:
10306:
10279:
10226:
10193:, for example the
10076:
10036:
9943:
9916:
9892:
9840:
9800:
9771:
9589:
9491:
9464:
9444:
9415:
9350:
9275:
9251:free abelian group
9237:simplicial complex
9227:algebraic topology
9192:
9080:
9044:
9024:
9004:
8952:
8896:
8852:
8832:
8799:
8763:
8727:
8691:
8616:
8590:
8564:
8489:
8469:
8430:
8410:
8383:
8363:
8343:
8316:nth homotopy group
8289:
8189:
8114:
8061:; then the groups
8043:
7978:free abelian group
7951:simplicial complex
7918:
7882:
7839:
7807:
7785:
7759:
7723:
7671:
7641:
7608:for a combination
7595:
7584:
7546:
7509:
7486:
7228:. The elements of
7205:
7054:
7011:
6964:
6892:
6846:
6793:
6728:
6655:
6612:
6565:
6491:
6455:
6419:
6328:
6292:
6093:boundary operators
6081:
6016:
5913:
5862:
5817:
5812:
5696:
5658:
5609:
5604:
5480:
5423:
5386:
5381:
5255:
5192:
5187:
5092:
5072:-dimensional ball
5058:
5005:
4985:A two-dimensional
4969:
4912:
4872:
4867:
4767:
4743:In general for an
4731:
4726:
4626:
4606:A two-dimensional
4585:
4528:
4502:
4480:
4475:
4371:
4351:A one-dimensional
4334:
4296:
4244:
4192:
4156:
4124:
4030:The homology of a
4012:
3992:
3956:
3920:
3892:
3856:
3836:
3816:
3796:
3729:
3690:
3670:
3650:
3604:
3576:
3555:. By contrast, no
3545:
3518:
3498:
3471:
3451:
3424:
3404:
3384:
3364:
3344:
3317:
3285:
3242:
3215:
3184:
3157:
3137:
3117:
3097:
3077:
3057:
3037:
3010:
2983:
2956:
2929:
2902:
2875:
2848:
2821:
2794:
2767:
2740:
2713:
2686:
2659:
2632:
2605:
2573:
2546:
2506:
2464:
2441:. In other words,
2431:
2411:
2298:
2278:
2188:
2161:
2085:
2056:
2030:
1998:
1958:
1923:
1896:
1861:
1841:
1782:
1646:
1626:
1570:
1543:
1516:
1453:
1433:
1374:
1354:
1306:
1279:
1241:
1214:
1187:
1156:De Rham cohomology
1082:derived categories
1070:abelian categories
997:
966:
946:
919:
892:
865:
743:
713:
693:
586:
556:
532:
474:
457:th homology group
447:
424:
212:
178:
148:
74:topological spaces
47:algebraic topology
14894:
14893:
14683:fundamental group
14432:978-0-19-851045-1
14398:978-0-387-97970-0
14355:Spanier, Edwin H.
14208:Eilenberg, Samuel
14155:, pp. 61β63.
14059:978-0-203-91266-9
13959:10.1137/130906556
13443:
13442:
12634:Euclidean 3-space
12509:fundamental group
12358:Cycles on a Klein
12251:'s definition of
12173:Smith normal form
12124:established that
12102:dynamical systems
12076:algebraic variety
11642:relative homology
11009:cohomology groups
10666:Homology functors
10659:Steenrod homology
10644:Khovanov homology
10614:Cellular homology
10555:and have a given
10133:topological space
10129:singular homology
10123:Singular homology
10117:Singular homology
10111:Smith normal form
10007:
9863:
9467:{\displaystyle i}
9206:Types of homology
9047:{\displaystyle r}
9027:{\displaystyle l}
8868:fundamental group
8855:{\displaystyle X}
8842:example, suppose
8492:{\displaystyle X}
8442:fundamental group
8433:{\displaystyle X}
8386:{\displaystyle n}
8366:{\displaystyle X}
7998:singular homology
7980:generated by the
7705:
7575:
7537:
7470:
7447:
7417:
7394:
7352:
7322:
6856:is a subgroup of
6287:
6257:
6227:
6204:
6162:
6132:
6091:which are called
5808:
5600:
5556:
5377:
5183:
4863:
4722:
4471:
4032:topological space
4026:Informal examples
4015:{\displaystyle n}
3923:{\displaystyle n}
3859:{\displaystyle n}
3839:{\displaystyle n}
3819:{\displaystyle n}
3693:{\displaystyle n}
3673:{\displaystyle X}
3607:{\displaystyle X}
3521:{\displaystyle C}
3474:{\displaystyle D}
3427:{\displaystyle C}
3407:{\displaystyle X}
3387:{\displaystyle X}
3367:{\displaystyle X}
3347:{\displaystyle X}
3160:{\displaystyle Y}
3140:{\displaystyle Y}
3120:{\displaystyle Y}
3100:{\displaystyle Y}
3080:{\displaystyle Y}
3060:{\displaystyle Y}
3040:{\displaystyle Y}
2434:{\displaystyle n}
2301:{\displaystyle k}
1864:{\displaystyle n}
1649:{\displaystyle n}
1456:{\displaystyle X}
1377:{\displaystyle X}
1137:cellular homology
1133:singular homology
1110:singular homology
1038:Khovanov homology
1030:singular homology
1019:Homology Theories
969:{\displaystyle R}
922:{\displaystyle R}
716:{\displaystyle n}
559:{\displaystyle n}
450:{\displaystyle n}
379:
334:
301:
94:topological space
70:homology theories
16:(Redirected from
14914:
14884:
14883:
14857:
14856:
14847:
14837:
14827:
14826:
14815:
14814:
14609:
14522:
14515:
14508:
14499:
14498:
14468:
14453:
14443:
14401:
14371:
14349:
14337:
14308:
14287:
14260:
14233:
14190:
14184:
14178:
14162:
14156:
14145:
14139:
14129:
14123:
14117:
14111:
14105:
14099:
14093:
14087:
14082:
14076:
14070:
14064:
14063:
14043:
14034:
14028:
14022:
14016:
14010:
14009:
13969:
13963:
13962:
13952:
13926:
13917:
13908:
13907:
13897:
13880:(5): 1663β1721.
13871:
13862:
13856:
13855:
13853:
13851:
13841:
13835:
13834:
13832:
13830:
13821:. Archived from
13815:
13809:
13803:
13797:
13791:
13785:
13779:
13773:
13767:
13761:
13755:
13749:
13748:
13730:
13724:
13718:
13709:
13708:
13690:
13684:
13678:
13667:
13661:
13645:PoincarΓ© duality
13578:
13573:
13572:
13555:
13544:
13509:Leopold Vietoris
13490:
13488:
13487:
13482:
13480:
13479:
13432:
13428:
13421:
13417:
13398:
13394:
13387:
13383:
13379:
13373:
13369:
13362:
13355:
13349:2-Manifold with
13320:
13280:
13216:
13206:
13204:
13203:
13198:
13196:
13195:
13160:
13154:Projective plane
13150:
13148:
13147:
13142:
13140:
13139:
13104:
13094:
13092:
13091:
13086:
13084:
13083:
13048:
13038:
13036:
13035:
13030:
13028:
13027:
12992:
12981:
12979:
12978:
12973:
12971:
12970:
12919:(1-dimensional)
12895:
12891:
12696:
12694:
12693:
12688:
12686:
12685:
12669:projective plane
12627:
12625:
12624:
12619:
12617:
12616:
12543:
12541:
12540:
12535:
12533:
12532:
12506:
12504:
12503:
12498:
12496:
12495:
12460:
12458:
12457:
12452:
12450:
12449:
12431:On the ordinary
12424:
12422:
12421:
12416:
12414:
12413:
12395:
12386:
12384:
12383:
12378:
12376:
12375:
12355:
12346:
12344:
12343:
12338:
12336:
12335:
12317:
12308:
12306:
12305:
12300:
12298:
12297:
12279:
12153:cohomology class
12135:In one class of
12095:network topology
12057:
12055:
12054:
12049:
12028:
12026:
12025:
12020:
12018:
12017:
12012:
11993:
11991:
11990:
11985:
11983:
11982:
11977:
11954:antipodal points
11925:
11923:
11922:
11917:
11900:-sphere for any
11872:
11870:
11869:
11864:
11830:
11828:
11827:
11822:
11820:
11819:
11814:
11789:
11787:
11786:
11781:
11779:
11778:
11773:
11747:
11745:
11744:
11739:
11709:
11707:
11706:
11701:
11699:
11698:
11631:
11629:
11628:
11623:
11612:
11611:
11584:
11583:
11564:
11562:
11561:
11556:
11539:
11538:
11511:
11510:
11483:
11482:
11455:
11454:
11427:
11426:
11405:
11404:
11383:
11382:
11350:
11348:
11347:
11342:
11293:
11291:
11290:
11285:
11270:
11268:
11267:
11262:
11257:
11256:
11244:
11229:
11228:
11180:
11178:
11177:
11172:
11167:
11166:
11154:
11139:
11138:
11089:
11087:
11086:
11081:
11079:
11078:
11060:
11059:
11047:
11046:
10968:
10966:
10965:
10960:
10916:
10914:
10913:
10908:
10906:
10905:
10893:
10892:
10880:
10879:
10867:
10866:
10844:
10842:
10841:
10836:
10834:
10833:
10821:
10820:
10808:
10807:
10791:
10789:
10788:
10783:
10781:
10780:
10762:
10761:
10749:
10748:
10732:
10730:
10729:
10724:
10722:
10721:
10703:
10702:
10690:
10689:
10598:
10596:
10595:
10590:
10582:
10538:extension groups
10535:
10533:
10532:
10527:
10510:
10509:
10470:
10468:
10467:
10462:
10460:
10459:
10439:
10437:
10436:
10431:
10429:
10428:
10412:
10410:
10409:
10404:
10402:
10401:
10385:
10383:
10382:
10377:
10372:
10368:
10367:
10345:
10344:
10332:
10331:
10315:
10313:
10312:
10307:
10305:
10304:
10288:
10286:
10285:
10280:
10269:
10268:
10256:
10255:
10235:
10233:
10232:
10227:
10225:
10224:
10201:and some module
10191:derived functors
10187:abstract algebra
10181:Group cohomology
10093:-th homology of
10085:
10083:
10082:
10077:
10069:
10068:
10056:
10055:
10046:
10045:
10035:
10034:
10033:
10021:
10020:
9994:
9993:
9952:
9950:
9949:
9944:
9942:
9941:
9925:
9923:
9922:
9917:
9912:
9911:
9902:
9901:
9891:
9890:
9889:
9877:
9876:
9849:
9847:
9846:
9841:
9839:
9838:
9809:
9807:
9806:
9801:
9780:
9778:
9777:
9772:
9767:
9763:
9676:
9675:
9656:
9651:
9624:
9623:
9598:
9596:
9595:
9590:
9517:boundary mapping
9500:
9498:
9497:
9492:
9490:
9489:
9473:
9471:
9470:
9465:
9453:
9451:
9450:
9445:
9424:
9422:
9421:
9416:
9359:
9357:
9356:
9351:
9284:
9282:
9281:
9276:
9201:
9199:
9198:
9193:
9188:
9180:
9179:
9167:
9166:
9156:
9147:
9146:
9131:
9130:
9106:
9105:
9089:
9087:
9086:
9081:
9070:
9069:
9053:
9051:
9050:
9045:
9033:
9031:
9030:
9025:
9013:
9011:
9010:
9005:
9003:
8995:
8978:
8977:
8961:
8959:
8958:
8953:
8951:
8943:
8926:
8925:
8905:
8903:
8902:
8897:
8886:
8885:
8861:
8859:
8858:
8853:
8841:
8839:
8838:
8833:
8808:
8806:
8805:
8800:
8789:
8788:
8772:
8770:
8769:
8764:
8753:
8752:
8736:
8734:
8733:
8728:
8717:
8716:
8700:
8698:
8697:
8692:
8681:
8680:
8659:
8658:
8646:
8645:
8625:
8623:
8622:
8617:
8599:
8597:
8596:
8591:
8573:
8571:
8570:
8565:
8554:
8553:
8532:
8531:
8519:
8518:
8501:Hurewicz theorem
8498:
8496:
8495:
8490:
8479:. For connected
8478:
8476:
8475:
8470:
8459:
8458:
8439:
8437:
8436:
8431:
8419:
8417:
8416:
8411:
8409:
8408:
8392:
8390:
8389:
8384:
8372:
8370:
8369:
8364:
8352:
8350:
8349:
8344:
8333:
8332:
8298:
8296:
8295:
8290:
8276:
8275:
8266:
8252:
8251:
8230:
8229:
8198:
8196:
8195:
8190:
8179:
8178:
8166:
8162:
8161:
8142:
8123:
8121:
8120:
8115:
8104:
8103:
8091:
8087:
8086:
8052:
8050:
8049:
8044:
8039:
8038:
7927:
7925:
7924:
7919:
7908:
7907:
7891:
7889:
7888:
7883:
7872:
7871:
7848:
7846:
7845:
7840:
7816:
7814:
7813:
7808:
7806:
7794:
7792:
7791:
7786:
7768:
7766:
7765:
7760:
7749:
7748:
7732:
7730:
7729:
7724:
7713:
7712:
7707:
7706:
7698:
7680:
7678:
7677:
7672:
7667:
7666:
7650:
7648:
7647:
7642:
7637:
7636:
7627:
7626:
7604:
7602:
7601:
7596:
7594:
7593:
7583:
7571:
7567:
7566:
7565:
7556:
7555:
7545:
7518:
7516:
7515:
7510:
7495:
7493:
7492:
7487:
7482:
7476:
7471:
7466:
7460:
7458:
7457:
7448:
7446:
7445:
7436:
7430:
7428:
7427:
7418:
7416:
7415:
7406:
7400:
7395:
7393:
7392:
7377:
7371:
7369:
7368:
7353:
7351:
7350:
7341:
7335:
7333:
7332:
7323:
7321:
7320:
7305:
7299:
7241:homology classes
7214:
7212:
7211:
7206:
7192:
7191:
7182:
7168:
7167:
7152:
7151:
7133:
7125:
7117:
7116:
7086:
7085:
7063:
7061:
7060:
7055:
7050:
7049:
7020:
7018:
7017:
7012:
7007:
7006:
6988:
6973:
6971:
6970:
6965:
6960:
6959:
6935:
6934:
6916:
6901:
6899:
6898:
6893:
6888:
6887:
6855:
6853:
6852:
6847:
6842:
6841:
6802:
6800:
6799:
6794:
6789:
6788:
6758:
6757:
6742:and elements of
6737:
6735:
6734:
6729:
6724:
6723:
6705:
6685:
6684:
6664:
6662:
6661:
6656:
6651:
6650:
6621:
6619:
6618:
6613:
6608:
6607:
6589:
6574:
6572:
6571:
6566:
6561:
6560:
6536:
6535:
6517:
6500:
6498:
6497:
6492:
6487:
6486:
6464:
6462:
6461:
6456:
6454:
6453:
6428:
6426:
6425:
6420:
6415:
6414:
6384:
6383:
6365:
6364:
6337:
6335:
6334:
6329:
6321:
6320:
6301:
6299:
6298:
6293:
6288:
6286:
6285:
6276:
6270:
6268:
6267:
6258:
6256:
6255:
6246:
6240:
6238:
6237:
6228:
6226:
6225:
6216:
6210:
6205:
6203:
6202:
6187:
6181:
6179:
6178:
6163:
6161:
6160:
6151:
6145:
6143:
6142:
6133:
6131:
6130:
6115:
6109:
6090:
6088:
6087:
6082:
6077:
6076:
6058:
6057:
6045:
6044:
6025:
6023:
6022:
6017:
6009:
6008:
5996:
5995:
5983:
5982:
5922:
5920:
5919:
5914:
5912:
5911:
5906:
5888:
5887:
5871:
5869:
5868:
5863:
5861:
5844:
5843:
5826:
5824:
5823:
5818:
5816:
5815:
5809:
5806:
5778:
5777:
5772:
5751:
5726:
5725:
5705:
5703:
5702:
5697:
5695:
5694:
5689:
5672:projective plane
5667:
5665:
5664:
5659:
5654:
5646:
5618:
5616:
5615:
5610:
5608:
5607:
5601:
5598:
5564:
5563:
5562:
5561:
5548:
5541:
5521:
5520:
5508:
5507:
5489:
5487:
5486:
5481:
5479:
5478:
5469:
5468:
5453:
5452:
5432:
5430:
5429:
5424:
5422:
5421:
5395:
5393:
5392:
5387:
5385:
5384:
5378:
5375:
5347:
5339:
5313:
5294:
5293:
5281:
5280:
5264:
5262:
5261:
5256:
5254:
5253:
5241:
5240:
5228:
5227:
5208:is defined as a
5201:
5199:
5198:
5193:
5191:
5190:
5184:
5181:
5153:
5137:
5133:
5132:
5119:
5118:
5101:
5099:
5098:
5093:
5088:
5087:
5067:
5065:
5064:
5059:
5057:
5049:
5045:
5044:
5031:
5030:
5014:
5012:
5011:
5006:
5004:
5003:
4978:
4976:
4975:
4970:
4968:
4967:
4955:
4954:
4930:
4921:
4919:
4918:
4913:
4911:
4910:
4892:
4881:
4879:
4878:
4873:
4871:
4870:
4864:
4861:
4827:
4811:
4807:
4806:
4793:
4792:
4776:
4774:
4773:
4768:
4763:
4762:
4740:
4738:
4737:
4732:
4730:
4729:
4723:
4720:
4686:
4670:
4666:
4665:
4652:
4651:
4635:
4633:
4632:
4627:
4625:
4624:
4599:, with a single
4594:
4592:
4591:
4586:
4584:
4576:
4572:
4571:
4558:
4557:
4537:
4535:
4534:
4529:
4511:
4509:
4508:
4503:
4501:
4489:
4487:
4486:
4481:
4479:
4478:
4472:
4469:
4435:
4419:
4415:
4414:
4401:
4400:
4380:
4378:
4377:
4372:
4370:
4369:
4343:
4341:
4340:
4335:
4333:
4332:
4314:
4305:
4303:
4302:
4297:
4295:
4294:
4276:
4253:
4251:
4250:
4245:
4234:
4233:
4218:. Consequently,
4201:
4199:
4198:
4193:
4182:
4181:
4165:
4163:
4162:
4157:
4155:
4154:
4153:
4133:
4131:
4130:
4125:
4108:
4107:
4086:
4085:
4064:
4063:
4021:
4019:
4018:
4013:
4001:
3999:
3998:
3993:
3982:
3981:
3965:
3963:
3962:
3957:
3946:
3945:
3929:
3927:
3926:
3921:
3904:homology classes
3901:
3899:
3898:
3893:
3882:
3881:
3865:
3863:
3862:
3857:
3845:
3843:
3842:
3837:
3825:
3823:
3822:
3817:
3805:
3803:
3802:
3797:
3795:
3794:
3785:
3780:
3779:
3758:
3757:
3738:
3736:
3735:
3730:
3719:
3718:
3699:
3697:
3696:
3691:
3679:
3677:
3676:
3671:
3659:
3657:
3656:
3651:
3646:
3645:
3633:
3632:
3613:
3611:
3610:
3605:
3585:
3583:
3582:
3577:
3575:
3574:
3554:
3552:
3551:
3546:
3544:
3543:
3527:
3525:
3524:
3519:
3507:
3505:
3504:
3499:
3497:
3496:
3480:
3478:
3477:
3472:
3460:
3458:
3457:
3452:
3450:
3449:
3433:
3431:
3430:
3425:
3413:
3411:
3410:
3405:
3393:
3391:
3390:
3385:
3373:
3371:
3370:
3365:
3353:
3351:
3350:
3345:
3327:have nontrivial
3326:
3324:
3323:
3318:
3316:
3315:
3310:
3294:
3292:
3291:
3286:
3284:
3283:
3278:
3262:projective plane
3251:
3249:
3248:
3243:
3241:
3240:
3224:
3222:
3221:
3216:
3214:
3213:
3193:
3191:
3190:
3185:
3183:
3182:
3166:
3164:
3163:
3158:
3146:
3144:
3143:
3138:
3126:
3124:
3123:
3118:
3106:
3104:
3103:
3098:
3086:
3084:
3083:
3078:
3066:
3064:
3063:
3058:
3046:
3044:
3043:
3038:
3019:
3017:
3016:
3011:
3009:
3008:
2992:
2990:
2989:
2984:
2982:
2981:
2965:
2963:
2962:
2957:
2955:
2954:
2938:
2936:
2935:
2930:
2928:
2927:
2911:
2909:
2908:
2903:
2901:
2900:
2884:
2882:
2881:
2876:
2874:
2873:
2857:
2855:
2854:
2849:
2847:
2846:
2830:
2828:
2827:
2822:
2820:
2819:
2803:
2801:
2800:
2795:
2793:
2792:
2776:
2774:
2773:
2768:
2766:
2765:
2749:
2747:
2746:
2741:
2739:
2738:
2722:
2720:
2719:
2714:
2712:
2711:
2695:
2693:
2692:
2687:
2685:
2684:
2668:
2666:
2665:
2660:
2658:
2657:
2641:
2639:
2638:
2633:
2631:
2630:
2614:
2612:
2611:
2606:
2604:
2603:
2582:
2580:
2579:
2574:
2572:
2571:
2555:
2553:
2552:
2547:
2545:
2544:
2532:
2531:
2515:
2513:
2512:
2507:
2490:
2489:
2473:
2471:
2470:
2465:
2463:
2462:
2440:
2438:
2437:
2432:
2420:
2418:
2417:
2412:
2392:
2391:
2381:
2375:
2374:
2353:
2352:
2334:
2333:
2307:
2305:
2304:
2299:
2288:for any integer
2287:
2285:
2284:
2279:
2214:
2213:
2197:
2195:
2194:
2189:
2187:
2186:
2170:
2168:
2167:
2162:
2115:
2114:
2094:
2092:
2091:
2086:
2065:
2063:
2062:
2057:
2039:
2037:
2036:
2033:{\displaystyle }
2031:
2007:
2005:
2004:
1999:
1967:
1965:
1964:
1961:{\displaystyle }
1959:
1932:
1930:
1929:
1924:
1922:
1921:
1905:
1903:
1902:
1897:
1895:
1894:
1870:
1868:
1867:
1862:
1850:
1848:
1847:
1842:
1840:
1839:
1821:
1820:
1808:
1807:
1791:
1789:
1788:
1783:
1778:
1777:
1755:
1740:
1739:
1718:
1717:
1701:
1690:
1689:
1655:
1653:
1652:
1647:
1635:
1633:
1632:
1627:
1619:
1618:
1606:
1605:
1579:
1577:
1576:
1571:
1569:
1568:
1552:
1550:
1549:
1544:
1542:
1541:
1525:
1523:
1522:
1517:
1515:
1514:
1496:
1495:
1483:
1482:
1462:
1460:
1459:
1454:
1442:
1440:
1439:
1434:
1429:
1428:
1416:
1415:
1383:
1381:
1380:
1375:
1363:
1361:
1360:
1355:
1344:
1343:
1318:formal addition,
1315:
1313:
1312:
1307:
1305:
1304:
1288:
1286:
1285:
1280:
1278:
1277:
1250:
1248:
1247:
1242:
1240:
1239:
1223:
1221:
1220:
1215:
1213:
1212:
1197:, and a 2-torus
1196:
1194:
1193:
1188:
1186:
1185:
1086:model categories
1066:derived functors
1009:quotient modules
1006:
1004:
1003:
998:
996:
995:
975:
973:
972:
967:
955:
953:
952:
947:
945:
944:
928:
926:
925:
920:
908:coefficient ring
901:
899:
898:
893:
891:
890:
874:
872:
871:
866:
861:
860:
838:
823:
822:
801:
800:
784:
773:
772:
753:is given by the
752:
750:
749:
744:
742:
741:
722:
720:
719:
714:
702:
700:
699:
694:
674:
673:
663:
657:
656:
635:
634:
616:
615:
596:is given by the
595:
593:
592:
587:
585:
584:
565:
563:
562:
557:
541:
539:
538:
533:
531:
530:
521:
516:
515:
503:
502:
483:
481:
480:
475:
473:
472:
456:
454:
453:
448:
433:
431:
430:
425:
417:
416:
398:
397:
381:
380:
378:
377:
376:
360:
355:
352:
351:
336:
335:
333:
332:
331:
321:
316:
313:
312:
303:
302:
300:
299:
298:
282:
277:
274:
273:
249:
248:
221:
219:
218:
213:
211:
210:
187:
185:
184:
179:
177:
176:
157:
155:
154:
149:
144:
143:
131:
130:
62:homology groups.
21:
14922:
14921:
14917:
14916:
14915:
14913:
14912:
14911:
14907:Homology theory
14897:
14896:
14895:
14890:
14821:
14803:
14799:Urysohn's lemma
14760:
14724:
14610:
14601:
14573:low-dimensional
14531:
14526:
14476:
14466:
14451:
14433:
14399:
14377:Stillwell, John
14369:
14306:
14285:
14265:Gowers, Timothy
14249:
14222:
14199:
14197:Further reading
14194:
14193:
14185:
14181:
14175:Wayback Machine
14163:
14159:
14146:
14142:
14130:
14126:
14118:
14114:
14106:
14102:
14094:
14090:
14083:
14079:
14071:
14067:
14060:
14044:
14037:
14029:
14025:
14017:
14013:
13970:
13966:
13950:10.1.1.716.3210
13924:
13918:
13911:
13869:
13863:
13859:
13849:
13847:
13843:
13842:
13838:
13828:
13826:
13825:on 22 June 2007
13817:
13816:
13812:
13804:
13800:
13792:
13788:
13780:
13776:
13768:
13764:
13756:
13752:
13731:
13727:
13719:
13712:
13691:
13687:
13679:
13670:
13662:
13658:
13653:
13634:KΓΌnneth theorem
13574:
13567:
13564:
13559:
13558:
13545:
13541:
13536:
13475:
13471:
13469:
13466:
13465:
13430:
13426:
13419:
13415:
13396:
13392:
13385:
13381:
13377:
13371:
13367:
13360:
13353:
13318:
13278:
13214:
13191:
13187:
13185:
13182:
13181:
13158:
13135:
13131:
13129:
13126:
13125:
13102:
13079:
13075:
13073:
13070:
13069:
13046:
13023:
13019:
13017:
13014:
13013:
12990:
12966:
12962:
12960:
12957:
12956:
12951:
12943:
12935:
12918:
12905:
12681:
12677:
12675:
12672:
12671:
12612:
12608:
12606:
12603:
12602:
12528:
12524:
12522:
12519:
12518:
12491:
12487:
12485:
12482:
12481:
12445:
12441:
12439:
12436:
12435:
12429:
12428:
12427:
12426:
12425:
12409:
12405:
12403:
12400:
12399:
12396:
12388:
12387:
12371:
12367:
12365:
12362:
12361:
12359:
12356:
12348:
12347:
12331:
12327:
12325:
12322:
12321:
12318:
12310:
12309:
12293:
12289:
12287:
12284:
12283:
12280:
12269:
12241:
12236:
12196:Wayback Machine
12161:
12126:periodic orbits
12091:sensor networks
12080:Euclidean space
12064:
12034:
12031:
12030:
12013:
12008:
12007:
11999:
11996:
11995:
11978:
11973:
11972:
11964:
11961:
11960:
11905:
11902:
11901:
11843:
11840:
11839:
11815:
11810:
11809:
11795:
11792:
11791:
11774:
11769:
11768:
11766:
11763:
11762:
11715:
11712:
11711:
11694:
11690:
11682:
11679:
11678:
11659:
11654:
11601:
11597:
11579:
11575:
11573:
11570:
11569:
11528:
11524:
11500:
11496:
11472:
11468:
11444:
11440:
11422:
11418:
11400:
11396:
11378:
11374:
11366:
11363:
11362:
11312:
11309:
11308:
11294:for the object
11279:
11276:
11275:
11252:
11248:
11231:
11224:
11220:
11200:
11197:
11196:
11190:Hamel dimension
11162:
11158:
11141:
11134:
11130:
11110:
11107:
11106:
11095:
11068:
11064:
11055:
11051:
11042:
11038:
11036:
11033:
11032:
11029:
10982:
10948:
10945:
10944:
10930:
10901:
10897:
10888:
10884:
10875:
10871:
10856:
10852:
10850:
10847:
10846:
10829:
10825:
10816:
10812:
10803:
10799:
10797:
10794:
10793:
10770:
10766:
10757:
10753:
10744:
10740:
10738:
10735:
10734:
10711:
10707:
10698:
10694:
10685:
10681:
10679:
10676:
10675:
10668:
10663:
10619:Cyclic homology
10604:
10578:
10567:
10564:
10563:
10553:normal subgroup
10505:
10501:
10499:
10496:
10495:
10455:
10451:
10449:
10446:
10445:
10424:
10420:
10418:
10415:
10414:
10397:
10393:
10391:
10388:
10387:
10363:
10359:
10355:
10340:
10336:
10327:
10323:
10321:
10318:
10317:
10300:
10296:
10294:
10291:
10290:
10264:
10260:
10251:
10247:
10245:
10242:
10241:
10220:
10216:
10214:
10211:
10210:
10183:
10177:
10170:
10147:
10125:
10119:
10064:
10060:
10051:
10047:
10041:
10037:
10029:
10025:
10016:
10012:
10011:
9989:
9985:
9983:
9980:
9979:
9973:
9966:
9937:
9933:
9931:
9928:
9927:
9907:
9903:
9897:
9893:
9885:
9881:
9872:
9868:
9867:
9855:
9852:
9851:
9834:
9830:
9822:
9819:
9818:
9815:
9789:
9786:
9785:
9681:
9677:
9671:
9667:
9652:
9641:
9619:
9615:
9613:
9610:
9609:
9527:
9524:
9523:
9513:
9506:
9485:
9481:
9479:
9476:
9475:
9459:
9456:
9455:
9430:
9427:
9426:
9365:
9362:
9361:
9294:
9291:
9290:
9270:
9267:
9266:
9247:
9223:
9217:
9208:
9184:
9175:
9171:
9162:
9158:
9152:
9142:
9138:
9126:
9122:
9101:
9097:
9095:
9092:
9091:
9065:
9061:
9059:
9056:
9055:
9039:
9036:
9035:
9019:
9016:
9015:
8999:
8991:
8973:
8969:
8967:
8964:
8963:
8947:
8939:
8921:
8917:
8915:
8912:
8911:
8881:
8877:
8875:
8872:
8871:
8847:
8844:
8843:
8821:
8818:
8817:
8784:
8780:
8778:
8775:
8774:
8748:
8744:
8742:
8739:
8738:
8712:
8708:
8706:
8703:
8702:
8676:
8672:
8654:
8650:
8641:
8637:
8635:
8632:
8631:
8605:
8602:
8601:
8579:
8576:
8575:
8549:
8545:
8527:
8523:
8514:
8510:
8508:
8505:
8504:
8484:
8481:
8480:
8454:
8450:
8448:
8445:
8444:
8425:
8422:
8421:
8404:
8400:
8398:
8395:
8394:
8378:
8375:
8374:
8358:
8355:
8354:
8328:
8324:
8322:
8319:
8318:
8312:
8271:
8267:
8262:
8247:
8243:
8225:
8221:
8219:
8216:
8215:
8174:
8170:
8151:
8147:
8143:
8135:
8133:
8130:
8129:
8099:
8095:
8082:
8078:
8074:
8066:
8063:
8062:
8034:
8030:
8028:
8025:
8024:
8021:cochain complex
8006:
7970:
7942:
7903:
7899:
7897:
7894:
7893:
7867:
7863:
7861:
7858:
7857:
7822:
7819:
7818:
7802:
7800:
7797:
7796:
7774:
7771:
7770:
7744:
7740:
7738:
7735:
7734:
7708:
7697:
7696:
7695:
7693:
7690:
7689:
7687:
7662:
7658:
7656:
7653:
7652:
7632:
7628:
7622:
7618:
7613:
7610:
7609:
7589:
7585:
7579:
7561:
7557:
7551:
7547:
7541:
7536:
7532:
7527:
7524:
7523:
7504:
7501:
7500:
7477:
7472:
7461:
7459:
7453:
7449:
7441:
7437:
7431:
7429:
7423:
7419:
7411:
7407:
7401:
7399:
7382:
7378:
7372:
7370:
7358:
7354:
7346:
7342:
7336:
7334:
7328:
7324:
7310:
7306:
7300:
7298:
7293:
7290:
7289:
7233:
7187:
7183:
7178:
7163:
7159:
7141:
7137:
7126:
7121:
7112:
7108:
7081:
7077:
7075:
7072:
7071:
7045:
7041:
7030:
7027:
7026:
7023:normal subgroup
6996:
6992:
6981:
6979:
6976:
6975:
6955:
6951:
6924:
6920:
6909:
6907:
6904:
6903:
6883:
6879:
6868:
6865:
6864:
6861:
6837:
6833:
6822:
6819:
6818:
6815:
6784:
6780:
6753:
6749:
6747:
6744:
6743:
6713:
6709:
6698:
6680:
6676:
6674:
6671:
6670:
6646:
6642:
6631:
6628:
6627:
6597:
6593:
6582:
6580:
6577:
6576:
6556:
6552:
6525:
6521:
6510:
6508:
6505:
6504:
6476:
6472:
6470:
6467:
6466:
6443:
6439:
6437:
6434:
6433:
6392:
6388:
6373:
6369:
6360:
6356:
6354:
6351:
6350:
6316:
6312:
6310:
6307:
6306:
6281:
6277:
6271:
6269:
6263:
6259:
6251:
6247:
6241:
6239:
6233:
6229:
6221:
6217:
6211:
6209:
6192:
6188:
6182:
6180:
6168:
6164:
6156:
6152:
6146:
6144:
6138:
6134:
6120:
6116:
6110:
6108:
6103:
6100:
6099:
6066:
6062:
6053:
6049:
6040:
6036:
6034:
6031:
6030:
6026:. connected by
6004:
6000:
5991:
5987:
5978:
5974:
5972:
5969:
5968:
5933:
5907:
5902:
5901:
5883:
5879:
5877:
5874:
5873:
5857:
5839:
5835:
5833:
5830:
5829:
5811:
5810:
5805:
5803:
5791:
5790:
5779:
5773:
5768:
5767:
5764:
5763:
5752:
5747:
5740:
5739:
5721:
5717:
5715:
5712:
5711:
5690:
5685:
5684:
5682:
5679:
5678:
5650:
5642:
5640:
5637:
5636:
5603:
5602:
5597:
5595:
5583:
5582:
5565:
5557:
5544:
5543:
5542:
5537:
5536:
5529:
5528:
5516:
5512:
5503:
5499:
5497:
5494:
5493:
5474:
5470:
5464:
5460:
5448:
5444:
5442:
5439:
5438:
5417:
5413:
5411:
5408:
5407:
5380:
5379:
5374:
5372:
5360:
5359:
5348:
5343:
5335:
5332:
5331:
5314:
5309:
5302:
5301:
5289:
5285:
5276:
5272:
5270:
5267:
5266:
5249:
5245:
5236:
5232:
5223:
5219:
5217:
5214:
5213:
5212:of two circles
5186:
5185:
5180:
5178:
5166:
5165:
5154:
5149:
5142:
5141:
5128:
5124:
5120:
5114:
5110:
5108:
5105:
5104:
5083:
5079:
5077:
5074:
5073:
5053:
5040:
5036:
5032:
5026:
5022:
5020:
5017:
5016:
4999:
4995:
4993:
4990:
4989:
4983:
4982:
4981:
4980:
4979:
4963:
4959:
4950:
4946:
4938:
4935:
4934:
4931:
4923:
4922:
4906:
4902:
4900:
4897:
4896:
4893:
4866:
4865:
4860:
4858:
4846:
4845:
4828:
4823:
4816:
4815:
4802:
4798:
4794:
4788:
4784:
4782:
4779:
4778:
4758:
4754:
4752:
4749:
4748:
4725:
4724:
4719:
4717:
4705:
4704:
4687:
4682:
4675:
4674:
4661:
4657:
4653:
4647:
4643:
4641:
4638:
4637:
4620:
4616:
4614:
4611:
4610:
4580:
4567:
4563:
4559:
4553:
4549:
4547:
4544:
4543:
4517:
4514:
4513:
4497:
4495:
4492:
4491:
4474:
4473:
4468:
4466:
4454:
4453:
4436:
4431:
4424:
4423:
4410:
4406:
4402:
4396:
4392:
4390:
4387:
4386:
4365:
4361:
4359:
4356:
4355:
4349:
4348:
4347:
4346:
4345:
4328:
4324:
4322:
4319:
4318:
4315:
4307:
4306:
4290:
4286:
4284:
4281:
4280:
4277:
4266:
4229:
4225:
4223:
4220:
4219:
4177:
4173:
4171:
4168:
4167:
4166:homology group
4146:
4145:
4141:
4139:
4136:
4135:
4103:
4099:
4081:
4077:
4059:
4055:
4053:
4050:
4049:
4047:homology groups
4028:
4007:
4004:
4003:
3977:
3973:
3971:
3968:
3967:
3941:
3937:
3935:
3932:
3931:
3915:
3912:
3911:
3877:
3873:
3871:
3868:
3867:
3851:
3848:
3847:
3831:
3828:
3827:
3811:
3808:
3807:
3790:
3786:
3781:
3775:
3771:
3753:
3749:
3747:
3744:
3743:
3714:
3710:
3708:
3705:
3704:
3685:
3682:
3681:
3665:
3662:
3661:
3641:
3637:
3628:
3624:
3619:
3616:
3615:
3599:
3596:
3595:
3592:
3590:Homology groups
3570:
3566:
3564:
3561:
3560:
3539:
3535:
3533:
3530:
3529:
3513:
3510:
3509:
3492:
3488:
3486:
3483:
3482:
3466:
3463:
3462:
3445:
3441:
3439:
3436:
3435:
3419:
3416:
3415:
3399:
3396:
3395:
3379:
3376:
3375:
3359:
3356:
3355:
3339:
3336:
3335:
3311:
3303:
3302:
3300:
3297:
3296:
3279:
3271:
3270:
3268:
3265:
3264:
3236:
3232:
3230:
3227:
3226:
3209:
3205:
3203:
3200:
3199:
3178:
3174:
3172:
3169:
3168:
3152:
3149:
3148:
3132:
3129:
3128:
3112:
3109:
3108:
3092:
3089:
3088:
3072:
3069:
3068:
3052:
3049:
3048:
3032:
3029:
3028:
3004:
3000:
2998:
2995:
2994:
2977:
2973:
2971:
2968:
2967:
2950:
2946:
2944:
2941:
2940:
2923:
2919:
2917:
2914:
2913:
2896:
2892:
2890:
2887:
2886:
2869:
2865:
2863:
2860:
2859:
2842:
2838:
2836:
2833:
2832:
2815:
2811:
2809:
2806:
2805:
2788:
2784:
2782:
2779:
2778:
2761:
2757:
2755:
2752:
2751:
2734:
2730:
2728:
2725:
2724:
2707:
2703:
2701:
2698:
2697:
2680:
2676:
2674:
2671:
2670:
2653:
2649:
2647:
2644:
2643:
2626:
2622:
2620:
2617:
2616:
2599:
2595:
2593:
2590:
2589:
2567:
2563:
2561:
2558:
2557:
2540:
2536:
2527:
2523:
2521:
2518:
2517:
2485:
2481:
2479:
2476:
2475:
2458:
2454:
2446:
2443:
2442:
2426:
2423:
2422:
2387:
2383:
2377:
2370:
2366:
2348:
2344:
2329:
2325:
2323:
2320:
2319:
2293:
2290:
2289:
2209:
2205:
2203:
2200:
2199:
2182:
2178:
2176:
2173:
2172:
2110:
2106:
2104:
2101:
2100:
2071:
2068:
2067:
2045:
2042:
2041:
2013:
2010:
2009:
1973:
1970:
1969:
1941:
1938:
1937:
1917:
1913:
1911:
1908:
1907:
1890:
1886:
1884:
1881:
1880:
1856:
1853:
1852:
1829:
1825:
1816:
1812:
1803:
1799:
1797:
1794:
1793:
1767:
1763:
1751:
1729:
1725:
1707:
1703:
1694:
1685:
1681:
1679:
1676:
1675:
1665:chain complexes
1641:
1638:
1637:
1614:
1610:
1595:
1591:
1589:
1586:
1585:
1564:
1560:
1558:
1555:
1554:
1537:
1533:
1531:
1528:
1527:
1504:
1500:
1491:
1487:
1478:
1474:
1472:
1469:
1468:
1448:
1445:
1444:
1424:
1420:
1411:
1407:
1402:
1399:
1398:
1369:
1366:
1365:
1339:
1335:
1333:
1330:
1329:
1326:
1300:
1296:
1294:
1291:
1290:
1273:
1269:
1267:
1264:
1263:
1235:
1231:
1229:
1226:
1225:
1208:
1204:
1202:
1199:
1198:
1181:
1177:
1175:
1172:
1171:
1164:
1152:Δech cohomology
1098:
1068:on appropriate
1053:category theory
1025:homology theory
1023:To associate a
1021:
991:
987:
985:
982:
981:
961:
958:
957:
940:
936:
934:
931:
930:
914:
911:
910:
886:
882:
880:
877:
876:
850:
846:
834:
812:
808:
790:
786:
777:
768:
764:
762:
759:
758:
737:
733:
731:
728:
727:
708:
705:
704:
669:
665:
659:
652:
648:
630:
626:
611:
607:
605:
602:
601:
580:
576:
574:
571:
570:
551:
548:
547:
526:
522:
517:
511:
507:
498:
494:
492:
489:
488:
468:
464:
462:
459:
458:
442:
439:
438:
406:
402:
393:
389:
366:
362:
361:
356:
354:
353:
341:
337:
327:
323:
322:
317:
315:
314:
308:
304:
288:
284:
283:
278:
276:
275:
263:
259:
244:
240:
238:
235:
234:
206:
202:
200:
197:
196:
172:
168:
166:
163:
162:
139:
135:
126:
122:
117:
114:
113:
102:
90:cochain complex
35:
28:
23:
22:
18:Homology groups
15:
12:
11:
5:
14920:
14910:
14909:
14892:
14891:
14889:
14888:
14878:
14877:
14876:
14871:
14866:
14851:
14841:
14831:
14819:
14808:
14805:
14804:
14802:
14801:
14796:
14791:
14786:
14781:
14776:
14770:
14768:
14762:
14761:
14759:
14758:
14753:
14748:
14746:Winding number
14743:
14738:
14732:
14730:
14726:
14725:
14723:
14722:
14717:
14712:
14707:
14702:
14697:
14692:
14687:
14686:
14685:
14680:
14678:homotopy group
14670:
14669:
14668:
14663:
14658:
14653:
14648:
14638:
14633:
14628:
14618:
14616:
14612:
14611:
14604:
14602:
14600:
14599:
14594:
14589:
14588:
14587:
14577:
14576:
14575:
14565:
14560:
14555:
14550:
14545:
14539:
14537:
14533:
14532:
14525:
14524:
14517:
14510:
14502:
14496:
14495:
14490:
14485:
14481:Homology group
14475:
14474:External links
14472:
14471:
14470:
14464:
14444:
14431:
14409:, ed. (1999),
14403:
14397:
14373:
14367:
14351:
14338:
14310:
14304:
14289:
14283:
14261:
14247:
14234:
14220:
14198:
14195:
14192:
14191:
14179:
14157:
14140:
14124:
14112:
14100:
14088:
14077:
14065:
14058:
14035:
14023:
14019:Stillwell 1993
14011:
13964:
13909:
13857:
13836:
13810:
13808:, p. 126.
13798:
13786:
13774:
13762:
13750:
13725:
13710:
13685:
13668:
13655:
13654:
13652:
13649:
13648:
13647:
13642:
13636:
13631:
13629:Homotopy group
13626:
13621:
13616:
13611:
13606:
13601:
13596:
13591:
13586:
13580:
13579:
13563:
13560:
13557:
13556:
13538:
13537:
13535:
13532:
13521:abelian groups
13495:
13494:
13493:
13492:
13478:
13474:
13450:
13441:
13440:
13437:
13434:
13409:
13406:
13405:Non-orientable
13403:
13375:
13347:
13344:
13343:
13340:
13337:
13331:
13328:
13327:Non-orientable
13325:
13316:
13307:
13304:
13303:
13300:
13297:
13291:
13288:
13285:
13276:
13267:-holed torus (
13262:
13259:
13258:
13255:
13252:
13249:
13246:
13243:
13240:
13237:
13234:
13233:
13230:
13227:
13224:
13221:
13220:Non-orientable
13218:
13212:
13207:
13194:
13190:
13178:
13177:
13174:
13171:
13168:
13165:
13164:Non-orientable
13162:
13156:
13151:
13138:
13134:
13122:
13121:
13118:
13115:
13112:
13109:
13106:
13100:
13095:
13082:
13078:
13066:
13065:
13062:
13059:
13056:
13053:
13050:
13044:
13039:
13026:
13022:
13010:
13009:
13006:
13003:
13000:
12997:
12994:
12988:
12982:
12969:
12965:
12953:
12952:
12949:
12944:
12941:
12936:
12933:
12928:
12925:
12921:
12920:
12915:
12910:
12909:Orientability
12907:
12899:
12872:Analysis situs
12868:Henri PoincarΓ©
12847:A square is a
12684:
12680:
12615:
12611:
12531:
12527:
12494:
12490:
12448:
12444:
12412:
12408:
12397:
12390:
12389:
12374:
12370:
12357:
12350:
12349:
12334:
12330:
12319:
12312:
12311:
12296:
12292:
12281:
12274:
12273:
12272:
12271:
12270:
12268:
12265:
12240:
12237:
12235:
12232:
12160:
12157:
12118:Floer homology
12063:
12060:
12059:
12058:
12047:
12044:
12041:
12038:
12016:
12011:
12006:
12003:
11981:
11976:
11971:
11968:
11957:
11927:
11915:
11912:
11909:
11890:
11862:
11859:
11856:
11853:
11850:
11847:
11836:continuous map
11818:
11813:
11808:
11805:
11802:
11799:
11777:
11772:
11748:
11737:
11734:
11731:
11728:
11725:
11722:
11719:
11697:
11693:
11689:
11686:
11658:
11655:
11653:
11650:
11647:
11635:
11621:
11618:
11615:
11610:
11607:
11604:
11600:
11596:
11593:
11590:
11587:
11582:
11578:
11566:
11565:
11554:
11551:
11548:
11545:
11542:
11537:
11534:
11531:
11527:
11523:
11520:
11517:
11514:
11509:
11506:
11503:
11499:
11495:
11492:
11489:
11486:
11481:
11478:
11475:
11471:
11467:
11464:
11461:
11458:
11453:
11450:
11447:
11443:
11439:
11436:
11433:
11430:
11425:
11421:
11417:
11414:
11411:
11408:
11403:
11399:
11395:
11392:
11389:
11386:
11381:
11377:
11373:
11370:
11352:
11351:
11340:
11337:
11334:
11331:
11328:
11325:
11322:
11319:
11316:
11283:
11272:
11271:
11260:
11255:
11251:
11247:
11243:
11240:
11237:
11234:
11227:
11223:
11219:
11216:
11213:
11210:
11207:
11204:
11182:
11181:
11170:
11165:
11161:
11157:
11153:
11150:
11147:
11144:
11137:
11133:
11129:
11126:
11123:
11120:
11117:
11114:
11093:
11077:
11074:
11071:
11067:
11063:
11058:
11054:
11050:
11045:
11041:
11028:
11025:
10984:are covariant
10980:
10958:
10955:
10952:
10928:
10904:
10900:
10896:
10891:
10887:
10883:
10878:
10874:
10870:
10865:
10862:
10859:
10855:
10832:
10828:
10824:
10819:
10815:
10811:
10806:
10802:
10779:
10776:
10773:
10769:
10765:
10760:
10756:
10752:
10747:
10743:
10720:
10717:
10714:
10710:
10706:
10701:
10697:
10693:
10688:
10684:
10667:
10664:
10662:
10661:
10656:
10651:
10649:Morse homology
10646:
10641:
10636:
10631:
10629:Floer homology
10626:
10621:
10616:
10611:
10605:
10603:
10600:
10588:
10585:
10581:
10577:
10574:
10571:
10557:quotient group
10525:
10522:
10519:
10516:
10513:
10508:
10504:
10458:
10454:
10427:
10423:
10400:
10396:
10375:
10371:
10366:
10362:
10358:
10354:
10351:
10348:
10343:
10339:
10335:
10330:
10326:
10303:
10299:
10278:
10275:
10272:
10267:
10263:
10259:
10254:
10250:
10223:
10219:
10179:Main article:
10176:
10175:Group homology
10173:
10166:
10145:
10121:Main article:
10118:
10115:
10087:
10086:
10075:
10072:
10067:
10063:
10059:
10054:
10050:
10044:
10040:
10032:
10028:
10024:
10019:
10015:
10010:
10006:
10003:
10000:
9997:
9992:
9988:
9971:
9964:
9957:-simplexes in
9953:is the set of
9940:
9936:
9915:
9910:
9906:
9900:
9896:
9888:
9884:
9880:
9875:
9871:
9866:
9862:
9859:
9837:
9833:
9829:
9826:
9813:
9799:
9796:
9793:
9782:
9781:
9770:
9766:
9762:
9759:
9756:
9753:
9750:
9747:
9744:
9741:
9738:
9735:
9732:
9729:
9726:
9723:
9720:
9717:
9714:
9711:
9708:
9705:
9702:
9699:
9696:
9693:
9690:
9687:
9684:
9680:
9674:
9670:
9666:
9663:
9660:
9655:
9650:
9647:
9644:
9640:
9636:
9633:
9630:
9627:
9622:
9618:
9600:
9599:
9588:
9585:
9582:
9579:
9576:
9573:
9570:
9567:
9564:
9561:
9558:
9555:
9552:
9549:
9546:
9543:
9540:
9537:
9534:
9531:
9518:
9515:is called the
9511:
9504:
9488:
9484:
9463:
9443:
9440:
9437:
9434:
9414:
9411:
9408:
9405:
9402:
9399:
9396:
9393:
9390:
9387:
9384:
9381:
9378:
9375:
9372:
9369:
9349:
9346:
9343:
9340:
9337:
9334:
9331:
9328:
9325:
9322:
9319:
9316:
9313:
9310:
9307:
9304:
9301:
9298:
9274:
9245:
9219:Main article:
9216:
9213:
9207:
9204:
9191:
9187:
9183:
9178:
9174:
9170:
9165:
9161:
9155:
9150:
9145:
9141:
9137:
9134:
9129:
9125:
9121:
9118:
9115:
9112:
9109:
9104:
9100:
9079:
9076:
9073:
9068:
9064:
9043:
9023:
9002:
8998:
8994:
8990:
8987:
8984:
8981:
8976:
8972:
8950:
8946:
8942:
8938:
8935:
8932:
8929:
8924:
8920:
8895:
8892:
8889:
8884:
8880:
8851:
8831:
8828:
8825:
8798:
8795:
8792:
8787:
8783:
8762:
8759:
8756:
8751:
8747:
8726:
8723:
8720:
8715:
8711:
8690:
8687:
8684:
8679:
8675:
8671:
8668:
8665:
8662:
8657:
8653:
8649:
8644:
8640:
8628:abelianization
8615:
8612:
8609:
8589:
8586:
8583:
8563:
8560:
8557:
8552:
8548:
8544:
8541:
8538:
8535:
8530:
8526:
8522:
8517:
8513:
8488:
8468:
8465:
8462:
8457:
8453:
8429:
8407:
8403:
8382:
8362:
8342:
8339:
8336:
8331:
8327:
8311:
8308:
8300:
8299:
8288:
8285:
8282:
8279:
8274:
8270:
8265:
8261:
8258:
8255:
8250:
8246:
8242:
8239:
8236:
8233:
8228:
8224:
8202:
8188:
8185:
8182:
8177:
8173:
8169:
8165:
8160:
8157:
8154:
8150:
8146:
8141:
8138:
8113:
8110:
8107:
8102:
8098:
8094:
8090:
8085:
8081:
8077:
8073:
8070:
8042:
8037:
8033:
8004:
7984:-simplices of
7968:
7940:
7917:
7914:
7911:
7906:
7902:
7881:
7878:
7875:
7870:
7866:
7838:
7835:
7832:
7829:
7826:
7805:
7784:
7781:
7778:
7758:
7755:
7752:
7747:
7743:
7733:coincide with
7722:
7719:
7716:
7711:
7704:
7701:
7685:
7670:
7665:
7661:
7640:
7635:
7631:
7625:
7621:
7617:
7606:
7605:
7592:
7588:
7582:
7578:
7574:
7570:
7564:
7560:
7554:
7550:
7544:
7540:
7535:
7531:
7508:
7497:
7496:
7485:
7480:
7475:
7469:
7464:
7456:
7452:
7444:
7440:
7434:
7426:
7422:
7414:
7410:
7404:
7398:
7391:
7388:
7385:
7381:
7375:
7367:
7364:
7361:
7357:
7349:
7345:
7339:
7331:
7327:
7319:
7316:
7313:
7309:
7303:
7297:
7231:
7216:
7215:
7204:
7201:
7198:
7195:
7190:
7186:
7181:
7177:
7174:
7171:
7166:
7162:
7158:
7155:
7150:
7147:
7144:
7140:
7136:
7132:
7129:
7124:
7120:
7115:
7111:
7107:
7104:
7101:
7098:
7095:
7092:
7089:
7084:
7080:
7066:quotient group
7053:
7048:
7044:
7040:
7037:
7034:
7010:
7005:
7002:
6999:
6995:
6991:
6987:
6984:
6963:
6958:
6954:
6950:
6947:
6944:
6941:
6938:
6933:
6930:
6927:
6923:
6919:
6915:
6912:
6891:
6886:
6882:
6878:
6875:
6872:
6859:
6845:
6840:
6836:
6832:
6829:
6826:
6813:
6792:
6787:
6783:
6779:
6776:
6773:
6770:
6767:
6764:
6761:
6756:
6752:
6727:
6722:
6719:
6716:
6712:
6708:
6704:
6701:
6697:
6694:
6691:
6688:
6683:
6679:
6669:. Elements of
6654:
6649:
6645:
6641:
6638:
6635:
6611:
6606:
6603:
6600:
6596:
6592:
6588:
6585:
6564:
6559:
6555:
6551:
6548:
6545:
6542:
6539:
6534:
6531:
6528:
6524:
6520:
6516:
6513:
6490:
6485:
6482:
6479:
6475:
6452:
6449:
6446:
6442:
6430:
6429:
6418:
6413:
6410:
6407:
6404:
6401:
6398:
6395:
6391:
6387:
6382:
6379:
6376:
6372:
6368:
6363:
6359:
6327:
6324:
6319:
6315:
6303:
6302:
6291:
6284:
6280:
6274:
6266:
6262:
6254:
6250:
6244:
6236:
6232:
6224:
6220:
6214:
6208:
6201:
6198:
6195:
6191:
6185:
6177:
6174:
6171:
6167:
6159:
6155:
6149:
6141:
6137:
6129:
6126:
6123:
6119:
6113:
6107:
6080:
6075:
6072:
6069:
6065:
6061:
6056:
6052:
6048:
6043:
6039:
6015:
6012:
6007:
6003:
5999:
5994:
5990:
5986:
5981:
5977:
5955:
5937:graph homology
5932:
5929:
5910:
5905:
5900:
5897:
5894:
5891:
5886:
5882:
5860:
5856:
5853:
5850:
5847:
5842:
5838:
5814:
5804:
5802:
5799:
5796:
5793:
5792:
5789:
5786:
5783:
5780:
5776:
5771:
5766:
5765:
5762:
5759:
5756:
5753:
5750:
5746:
5745:
5743:
5738:
5735:
5732:
5729:
5724:
5720:
5693:
5688:
5657:
5653:
5649:
5645:
5606:
5596:
5594:
5591:
5588:
5585:
5584:
5581:
5578:
5575:
5572:
5569:
5566:
5560:
5555:
5552:
5547:
5540:
5535:
5534:
5532:
5527:
5524:
5519:
5515:
5511:
5506:
5502:
5477:
5473:
5467:
5463:
5459:
5456:
5451:
5447:
5420:
5416:
5406:is written as
5383:
5373:
5371:
5368:
5365:
5362:
5361:
5358:
5355:
5352:
5349:
5346:
5342:
5338:
5334:
5333:
5330:
5327:
5324:
5321:
5318:
5315:
5312:
5308:
5307:
5305:
5300:
5297:
5292:
5288:
5284:
5279:
5275:
5252:
5248:
5244:
5239:
5235:
5231:
5226:
5222:
5189:
5179:
5177:
5174:
5171:
5168:
5167:
5164:
5161:
5158:
5155:
5152:
5148:
5147:
5145:
5140:
5136:
5131:
5127:
5123:
5117:
5113:
5091:
5086:
5082:
5056:
5052:
5048:
5043:
5039:
5035:
5029:
5025:
5002:
4998:
4966:
4962:
4958:
4953:
4949:
4945:
4942:
4932:
4925:
4924:
4909:
4905:
4894:
4887:
4886:
4885:
4884:
4883:
4869:
4859:
4857:
4854:
4851:
4848:
4847:
4844:
4841:
4838:
4835:
4832:
4829:
4826:
4822:
4821:
4819:
4814:
4810:
4805:
4801:
4797:
4791:
4787:
4766:
4761:
4757:
4728:
4718:
4716:
4713:
4710:
4707:
4706:
4703:
4700:
4697:
4694:
4691:
4688:
4685:
4681:
4680:
4678:
4673:
4669:
4664:
4660:
4656:
4650:
4646:
4623:
4619:
4583:
4579:
4575:
4570:
4566:
4562:
4556:
4552:
4527:
4524:
4521:
4500:
4477:
4467:
4465:
4462:
4459:
4456:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4434:
4430:
4429:
4427:
4422:
4418:
4413:
4409:
4405:
4399:
4395:
4368:
4364:
4331:
4327:
4316:
4309:
4308:
4293:
4289:
4278:
4271:
4270:
4269:
4268:
4267:
4263:graph homology
4243:
4240:
4237:
4232:
4228:
4191:
4188:
4185:
4180:
4176:
4152:
4149:
4144:
4123:
4120:
4117:
4114:
4111:
4106:
4102:
4098:
4095:
4092:
4089:
4084:
4080:
4076:
4073:
4070:
4067:
4062:
4058:
4027:
4024:
4011:
3991:
3988:
3985:
3980:
3976:
3955:
3952:
3949:
3944:
3940:
3919:
3891:
3888:
3885:
3880:
3876:
3855:
3835:
3815:
3793:
3789:
3784:
3778:
3774:
3770:
3767:
3764:
3761:
3756:
3752:
3741:quotient group
3728:
3725:
3722:
3717:
3713:
3689:
3669:
3660:associated to
3649:
3644:
3640:
3636:
3631:
3627:
3623:
3603:
3591:
3588:
3573:
3569:
3542:
3538:
3517:
3495:
3491:
3470:
3448:
3444:
3423:
3403:
3383:
3363:
3343:
3314:
3309:
3306:
3282:
3277:
3274:
3239:
3235:
3212:
3208:
3181:
3177:
3156:
3136:
3116:
3096:
3076:
3056:
3036:
3007:
3003:
2980:
2976:
2953:
2949:
2926:
2922:
2899:
2895:
2885:such that the
2872:
2868:
2845:
2841:
2818:
2814:
2791:
2787:
2764:
2760:
2750:such that the
2737:
2733:
2710:
2706:
2683:
2679:
2656:
2652:
2629:
2625:
2602:
2598:
2570:
2566:
2543:
2539:
2535:
2530:
2526:
2505:
2502:
2499:
2496:
2493:
2488:
2484:
2461:
2457:
2453:
2450:
2430:
2410:
2407:
2404:
2401:
2398:
2395:
2390:
2386:
2380:
2373:
2369:
2365:
2362:
2359:
2356:
2351:
2347:
2343:
2340:
2337:
2332:
2328:
2297:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2212:
2208:
2185:
2181:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2113:
2109:
2084:
2081:
2078:
2075:
2055:
2052:
2049:
2029:
2026:
2023:
2020:
2017:
1997:
1994:
1991:
1987:
1983:
1980:
1977:
1957:
1954:
1951:
1948:
1945:
1920:
1916:
1893:
1889:
1860:
1838:
1835:
1832:
1828:
1824:
1819:
1815:
1811:
1806:
1802:
1781:
1776:
1773:
1770:
1766:
1762:
1759:
1754:
1749:
1746:
1743:
1738:
1735:
1732:
1728:
1724:
1721:
1716:
1713:
1710:
1706:
1700:
1697:
1693:
1688:
1684:
1645:
1625:
1622:
1617:
1613:
1609:
1604:
1601:
1598:
1594:
1567:
1563:
1540:
1536:
1513:
1510:
1507:
1503:
1499:
1494:
1490:
1486:
1481:
1477:
1465:abelian groups
1452:
1443:associated to
1432:
1427:
1423:
1419:
1414:
1410:
1406:
1373:
1353:
1350:
1347:
1342:
1338:
1325:
1322:
1303:
1299:
1276:
1272:
1238:
1234:
1211:
1207:
1184:
1180:
1163:
1160:
1144:Morse homology
1117:graph homology
1097:
1094:
1046:group homology
1034:Morse homology
1020:
1017:
1007:that are also
994:
990:
965:
943:
939:
918:
889:
885:
864:
859:
856:
853:
849:
845:
842:
837:
832:
829:
826:
821:
818:
815:
811:
807:
804:
799:
796:
793:
789:
783:
780:
776:
771:
767:
740:
736:
712:
692:
689:
686:
683:
680:
677:
672:
668:
662:
655:
651:
647:
644:
641:
638:
633:
629:
625:
622:
619:
614:
610:
583:
579:
555:
529:
525:
520:
514:
510:
506:
501:
497:
486:quotient group
471:
467:
446:
435:
434:
423:
420:
415:
412:
409:
405:
401:
396:
392:
387:
384:
375:
372:
369:
365:
359:
350:
347:
344:
340:
330:
326:
320:
311:
307:
297:
294:
291:
287:
281:
272:
269:
266:
262:
258:
255:
252:
247:
243:
209:
205:
175:
171:
160:abelian groups
147:
142:
138:
134:
129:
125:
121:
110:chain complex,
101:
98:
58:abelian groups
51:homology of a
26:
9:
6:
4:
3:
2:
14919:
14908:
14905:
14904:
14902:
14887:
14879:
14875:
14872:
14870:
14867:
14865:
14862:
14861:
14860:
14852:
14850:
14846:
14842:
14840:
14836:
14832:
14830:
14825:
14820:
14818:
14810:
14809:
14806:
14800:
14797:
14795:
14792:
14790:
14787:
14785:
14782:
14780:
14777:
14775:
14772:
14771:
14769:
14767:
14763:
14757:
14756:Orientability
14754:
14752:
14749:
14747:
14744:
14742:
14739:
14737:
14734:
14733:
14731:
14727:
14721:
14718:
14716:
14713:
14711:
14708:
14706:
14703:
14701:
14698:
14696:
14693:
14691:
14688:
14684:
14681:
14679:
14676:
14675:
14674:
14671:
14667:
14664:
14662:
14659:
14657:
14654:
14652:
14649:
14647:
14644:
14643:
14642:
14639:
14637:
14634:
14632:
14629:
14627:
14623:
14620:
14619:
14617:
14613:
14608:
14598:
14595:
14593:
14592:Set-theoretic
14590:
14586:
14583:
14582:
14581:
14578:
14574:
14571:
14570:
14569:
14566:
14564:
14561:
14559:
14556:
14554:
14553:Combinatorial
14551:
14549:
14546:
14544:
14541:
14540:
14538:
14534:
14530:
14523:
14518:
14516:
14511:
14509:
14504:
14503:
14500:
14493:
14491:
14488:
14486:
14484:
14482:
14478:
14477:
14467:
14465:9780080534077
14461:
14457:
14450:
14445:
14442:
14438:
14434:
14428:
14424:
14420:
14416:
14412:
14408:
14404:
14400:
14394:
14390:
14386:
14382:
14378:
14374:
14370:
14368:0-387-90646-0
14364:
14360:
14356:
14352:
14347:
14343:
14339:
14336:
14332:
14328:
14324:
14320:
14316:
14311:
14307:
14305:0-521-79540-0
14301:
14297:
14296:
14290:
14286:
14284:9781400830398
14280:
14276:
14275:
14270:
14266:
14262:
14258:
14254:
14250:
14248:9780821812556
14244:
14240:
14235:
14231:
14227:
14223:
14221:9780674079779
14217:
14213:
14209:
14205:
14201:
14200:
14189:, p. 264
14188:
14187:Richeson 2008
14183:
14176:
14172:
14169:
14167:
14161:
14154:
14150:
14144:
14137:
14135:
14128:
14122:, p. 284
14121:
14116:
14109:
14104:
14098:, p. 258
14097:
14096:Richeson 2008
14092:
14086:
14085:Richeson 2008
14081:
14075:, p. 254
14074:
14073:Richeson 2008
14069:
14061:
14055:
14052:. CRC Press.
14051:
14050:
14042:
14040:
14032:
14027:
14021:, p. 170
14020:
14015:
14007:
14003:
13999:
13995:
13991:
13987:
13983:
13979:
13978:Acta Numerica
13975:
13968:
13960:
13956:
13951:
13946:
13942:
13938:
13934:
13930:
13923:
13916:
13914:
13905:
13901:
13896:
13891:
13887:
13883:
13879:
13875:
13868:
13861:
13846:
13840:
13824:
13820:
13814:
13807:
13802:
13796:, p. 156
13795:
13790:
13784:, p. 110
13783:
13778:
13772:, p. 113
13771:
13766:
13759:
13754:
13746:
13742:
13741:
13736:
13729:
13723:, p. 106
13722:
13717:
13715:
13706:
13702:
13701:
13696:
13689:
13682:
13677:
13675:
13673:
13666:, p. 155
13665:
13660:
13656:
13646:
13643:
13640:
13637:
13635:
13632:
13630:
13627:
13625:
13622:
13620:
13617:
13615:
13612:
13610:
13607:
13605:
13602:
13600:
13597:
13595:
13592:
13590:
13587:
13585:
13582:
13581:
13577:
13571:
13566:
13553:
13549:
13546:in part from
13543:
13539:
13531:
13529:
13524:
13522:
13518:
13514:
13513:Walther Mayer
13510:
13506:
13502:
13500:
13499:chain complex
13476:
13472:
13463:
13459:
13455:
13454:connected sum
13451:
13448:
13447:
13445:
13444:
13438:
13435:
13424:
13414:
13410:
13407:
13404:
13401:
13391:
13376:
13366:
13359:
13352:
13348:
13346:
13345:
13341:
13338:
13335:
13332:
13329:
13326:
13324:
13317:
13315:
13312:
13308:
13306:
13305:
13301:
13298:
13296:
13292:
13289:
13286:
13284:
13277:
13274:
13270:
13266:
13263:
13261:
13260:
13256:
13253:
13250:
13247:
13244:
13241:
13239:2-holed torus
13238:
13236:
13235:
13231:
13228:
13225:
13222:
13219:
13213:
13211:
13208:
13192:
13188:
13180:
13179:
13175:
13172:
13169:
13166:
13163:
13157:
13155:
13152:
13136:
13132:
13124:
13123:
13119:
13116:
13113:
13110:
13107:
13101:
13099:
13096:
13080:
13076:
13068:
13067:
13063:
13060:
13057:
13054:
13051:
13045:
13043:
13040:
13024:
13020:
13012:
13011:
13001:
12998:
12995:
12989:
12986:
12983:
12967:
12963:
12955:
12954:
12948:
12945:
12940:
12937:
12932:
12929:
12926:
12923:
12922:
12914:
12913:Betti numbers
12903:
12896:
12890:
12887:
12885:
12884:Betti numbers
12881:
12877:
12873:
12869:
12864:
12862:
12858:
12854:
12850:
12845:
12843:
12839:
12835:
12831:
12827:
12824:= 0. Because
12823:
12819:
12815:
12811:
12807:
12803:
12799:
12795:
12791:
12787:
12783:
12779:
12775:
12771:
12766:
12764:
12760:
12756:
12752:
12748:
12744:
12740:
12736:
12732:
12727:
12725:
12720:
12716:
12712:
12708:
12704:
12700:
12682:
12678:
12670:
12665:
12663:
12659:
12655:
12651:
12647:
12643:
12639:
12635:
12631:
12613:
12609:
12596:
12592:
12588:
12586:
12582:
12578:
12574:
12569:
12567:
12563:
12559:
12555:
12551:
12547:
12529:
12525:
12517:
12512:
12510:
12492:
12488:
12479:
12475:
12471:
12468:
12464:
12446:
12442:
12434:
12410:
12406:
12394:
12372:
12368:
12354:
12332:
12328:
12316:
12294:
12290:
12278:
12264:
12262:
12258:
12254:
12250:
12246:
12231:
12229:
12225:
12221:
12217:
12216:presentations
12213:
12209:
12205:
12201:
12197:
12193:
12190:
12186:
12182:
12178:
12174:
12170:
12166:
12156:
12154:
12150:
12146:
12142:
12138:
12133:
12131:
12127:
12123:
12119:
12115:
12111:
12107:
12103:
12098:
12096:
12092:
12087:
12085:
12081:
12077:
12073:
12069:
12045:
12042:
12039:
12036:
12014:
12004:
12001:
11979:
11969:
11966:
11958:
11955:
11951:
11949:
11943:
11941:
11936:
11932:
11928:
11913:
11910:
11907:
11899:
11895:
11891:
11888:
11884:
11880:
11879:homeomorphism
11876:
11857:
11851:
11848:
11845:
11837:
11834:
11816:
11803:
11800:
11797:
11775:
11760:
11756:
11752:
11749:
11735:
11732:
11729:
11723:
11717:
11695:
11691:
11687:
11684:
11676:
11672:
11668:
11664:
11663:
11662:
11649:
11645:
11643:
11639:
11638:zig-zag lemma
11633:
11616:
11608:
11605:
11602:
11598:
11588:
11580:
11576:
11552:
11543:
11535:
11532:
11529:
11525:
11515:
11507:
11504:
11501:
11497:
11487:
11479:
11476:
11473:
11469:
11459:
11451:
11448:
11445:
11441:
11431:
11423:
11419:
11409:
11401:
11397:
11387:
11379:
11375:
11368:
11361:
11360:
11359:
11357:
11338:
11332:
11326:
11320:
11314:
11307:
11306:
11305:
11304:
11299:
11297:
11281:
11253:
11249:
11225:
11217:
11214:
11208:
11205:
11202:
11195:
11194:
11193:
11191:
11187:
11163:
11159:
11135:
11127:
11124:
11118:
11115:
11112:
11105:
11104:
11103:
11102:
11101:
11096:
11075:
11072:
11069:
11065:
11056:
11052:
11048:
11043:
11039:
11024:
11022:
11018:
11017:contravariant
11014:
11010:
11006:
11002:
11001:contravariant
10998:
10993:
10991:
10987:
10983:
10976:
10972:
10956:
10950:
10942:
10937:
10935:
10931:
10925:-th homology
10924:
10920:
10902:
10898:
10894:
10889:
10885:
10881:
10876:
10872:
10868:
10863:
10860:
10857:
10853:
10830:
10826:
10817:
10813:
10809:
10804:
10800:
10777:
10774:
10771:
10767:
10758:
10754:
10750:
10745:
10741:
10718:
10715:
10712:
10708:
10699:
10695:
10691:
10686:
10682:
10673:
10660:
10657:
10655:
10652:
10650:
10647:
10645:
10642:
10640:
10637:
10635:
10632:
10630:
10627:
10625:
10622:
10620:
10617:
10615:
10612:
10610:
10607:
10606:
10599:
10586:
10583:
10579:
10575:
10572:
10569:
10561:
10558:
10554:
10550:
10546:
10542:
10539:
10520:
10517:
10514:
10506:
10502:
10492:
10490:
10487:, applied to
10486:
10482:
10478:
10474:
10456:
10452:
10443:
10425:
10421:
10398:
10394:
10373:
10369:
10364:
10360:
10356:
10352:
10349:
10341:
10337:
10333:
10328:
10324:
10301:
10297:
10276:
10273:
10265:
10261:
10257:
10252:
10248:
10240:homomorphism
10239:
10221:
10217:
10208:
10204:
10200:
10196:
10192:
10188:
10182:
10172:
10169:
10164:
10160:
10157:-dimensional
10156:
10152:
10148:
10141:
10137:
10134:
10130:
10124:
10114:
10112:
10108:
10104:
10101:at dimension
10100:
10096:
10092:
10073:
10065:
10061:
10052:
10042:
10038:
10030:
10026:
10022:
10017:
10013:
10008:
10004:
9998:
9990:
9978:
9977:
9976:
9974:
9967:
9960:
9956:
9938:
9934:
9913:
9908:
9904:
9898:
9894:
9886:
9882:
9878:
9873:
9869:
9864:
9860:
9857:
9835:
9831:
9827:
9824:
9816:
9797:
9794:
9791:
9768:
9764:
9757:
9751:
9748:
9745:
9742:
9736:
9733:
9730:
9724:
9721:
9715:
9712:
9709:
9703:
9700:
9697:
9694:
9688:
9682:
9678:
9672:
9664:
9661:
9653:
9648:
9645:
9642:
9638:
9634:
9628:
9620:
9608:
9607:
9606:
9605:
9580:
9574:
9571:
9568:
9565:
9559:
9553:
9550:
9544:
9538:
9532:
9529:
9522:
9521:
9520:
9516:
9514:
9507:
9486:
9461:
9438:
9432:
9409:
9403:
9400:
9397:
9394:
9388:
9382:
9379:
9373:
9367:
9341:
9335:
9332:
9329:
9326:
9320:
9314:
9311:
9305:
9299:
9288:
9272:
9264:
9260:
9256:
9252:
9248:
9241:
9238:
9234:
9233:
9228:
9222:
9212:
9203:
9181:
9176:
9172:
9168:
9163:
9159:
9148:
9143:
9139:
9135:
9132:
9127:
9123:
9116:
9110:
9102:
9098:
9074:
9066:
9062:
9041:
9021:
8996:
8988:
8982:
8974:
8970:
8944:
8936:
8930:
8922:
8918:
8909:
8890:
8882:
8878:
8869:
8865:
8849:
8829:
8826:
8823:
8814:
8812:
8793:
8785:
8781:
8757:
8749:
8745:
8721:
8713:
8709:
8685:
8677:
8673:
8663:
8655:
8651:
8647:
8642:
8638:
8629:
8613:
8610:
8607:
8587:
8584:
8581:
8558:
8550:
8546:
8536:
8528:
8524:
8520:
8515:
8511:
8502:
8486:
8463:
8455:
8451:
8443:
8427:
8405:
8401:
8380:
8360:
8337:
8329:
8325:
8317:
8307:
8305:
8286:
8280:
8272:
8268:
8263:
8256:
8248:
8244:
8240:
8234:
8226:
8222:
8214:
8213:
8212:
8210:
8206:
8200:
8183:
8175:
8171:
8167:
8163:
8158:
8155:
8152:
8148:
8144:
8127:
8108:
8100:
8096:
8092:
8088:
8083:
8079:
8075:
8071:
8068:
8060:
8056:
8040:
8035:
8031:
8022:
8017:
8015:
8011:
8007:
8000:
7999:
7993:
7992:for details.
7991:
7987:
7983:
7979:
7975:
7971:
7964:
7960:
7956:
7953:
7952:
7947:
7943:
7936:
7935:
7929:
7912:
7904:
7900:
7892:and boundary
7876:
7868:
7864:
7854:
7852:
7836:
7782:
7779:
7776:
7753:
7745:
7741:
7717:
7709:
7699:
7684:
7668:
7663:
7659:
7638:
7633:
7629:
7623:
7619:
7615:
7590:
7586:
7580:
7576:
7572:
7568:
7562:
7558:
7552:
7548:
7542:
7538:
7533:
7529:
7522:
7521:
7520:
7506:
7483:
7467:
7454:
7450:
7442:
7424:
7420:
7412:
7396:
7389:
7386:
7383:
7365:
7362:
7359:
7355:
7347:
7329:
7325:
7317:
7314:
7311:
7295:
7288:
7287:
7286:
7284:
7280:
7276:
7271:
7269:
7265:
7261:
7257:
7253:
7248:
7246:
7242:
7239:) are called
7238:
7234:
7227:
7226:
7222:
7202:
7196:
7188:
7184:
7179:
7172:
7164:
7160:
7156:
7148:
7145:
7142:
7122:
7113:
7102:
7099:
7096:
7090:
7082:
7078:
7070:
7069:
7068:
7067:
7046:
7035:
7032:
7024:
7003:
7000:
6997:
6956:
6945:
6942:
6939:
6931:
6928:
6925:
6884:
6873:
6870:
6862:
6838:
6827:
6824:
6816:
6808:
6806:
6785:
6774:
6771:
6768:
6762:
6754:
6750:
6741:
6720:
6717:
6714:
6695:
6689:
6681:
6677:
6668:
6647:
6636:
6633:
6625:
6604:
6601:
6598:
6557:
6546:
6543:
6540:
6532:
6529:
6526:
6501:
6488:
6483:
6480:
6477:
6473:
6450:
6447:
6444:
6440:
6416:
6411:
6408:
6405:
6402:
6399:
6396:
6393:
6389:
6385:
6380:
6377:
6374:
6366:
6361:
6349:
6348:
6347:
6345:
6341:
6325:
6322:
6317:
6313:
6289:
6282:
6264:
6260:
6252:
6234:
6230:
6222:
6206:
6199:
6196:
6193:
6175:
6172:
6169:
6165:
6157:
6139:
6135:
6127:
6124:
6121:
6105:
6098:
6097:
6096:
6094:
6078:
6073:
6070:
6067:
6063:
6054:
6050:
6046:
6041:
6029:
6028:homomorphisms
6013:
6010:
6005:
6001:
5997:
5992:
5988:
5984:
5979:
5975:
5966:
5962:
5958:
5954:
5953:chain complex
5951:
5949:
5944:
5942:
5938:
5928:
5926:
5908:
5898:
5892:
5884:
5880:
5854:
5848:
5840:
5836:
5827:
5797:
5787:
5784:
5781:
5774:
5760:
5757:
5754:
5741:
5736:
5730:
5722:
5718:
5710:of order 2):
5709:
5691:
5676:
5673:
5668:
5655:
5647:
5635:
5634:product group
5630:
5628:
5624:
5619:
5589:
5579:
5576:
5573:
5570:
5567:
5553:
5550:
5530:
5525:
5517:
5513:
5504:
5500:
5491:
5475:
5465:
5461:
5454:
5449:
5445:
5436:
5418:
5414:
5405:
5401:
5396:
5366:
5356:
5353:
5350:
5340:
5328:
5325:
5322:
5319:
5316:
5303:
5298:
5290:
5286:
5277:
5273:
5250:
5246:
5242:
5237:
5233:
5229:
5224:
5220:
5211:
5207:
5202:
5172:
5162:
5159:
5156:
5143:
5138:
5134:
5129:
5125:
5121:
5115:
5111:
5102:
5089:
5084:
5080:
5071:
5050:
5046:
5041:
5037:
5033:
5027:
5023:
5000:
4996:
4988:
4964:
4960:
4956:
4951:
4947:
4943:
4940:
4929:
4907:
4903:
4891:
4882:
4852:
4842:
4839:
4836:
4833:
4830:
4817:
4812:
4808:
4803:
4799:
4795:
4789:
4785:
4764:
4759:
4755:
4746:
4741:
4711:
4701:
4698:
4695:
4692:
4689:
4676:
4671:
4667:
4662:
4658:
4654:
4648:
4644:
4621:
4617:
4609:
4604:
4602:
4598:
4595:represents a
4577:
4573:
4568:
4564:
4560:
4554:
4550:
4541:
4540:trivial group
4522:
4460:
4450:
4447:
4444:
4441:
4438:
4425:
4420:
4416:
4411:
4407:
4403:
4397:
4393:
4384:
4366:
4362:
4354:
4329:
4325:
4317:The 2-sphere
4313:
4291:
4287:
4275:
4264:
4259:
4257:
4238:
4230:
4226:
4217:
4213:
4209:
4205:
4186:
4178:
4174:
4142:
4121:
4118:
4112:
4104:
4100:
4096:
4090:
4082:
4078:
4074:
4068:
4060:
4056:
4048:
4044:
4040:
4036:
4033:
4023:
4009:
3986:
3978:
3974:
3950:
3942:
3938:
3917:
3909:
3905:
3886:
3878:
3874:
3853:
3833:
3813:
3791:
3787:
3782:
3776:
3772:
3768:
3762:
3754:
3750:
3742:
3723:
3715:
3711:
3703:
3687:
3667:
3642:
3638:
3634:
3629:
3625:
3601:
3587:
3571:
3567:
3558:
3540:
3536:
3515:
3493:
3489:
3468:
3446:
3442:
3421:
3401:
3381:
3361:
3341:
3332:
3330:
3312:
3280:
3263:
3259:
3255:
3237:
3233:
3210:
3206:
3197:
3179:
3175:
3154:
3134:
3114:
3094:
3074:
3054:
3034:
3026:
3021:
3005:
3001:
2978:
2974:
2951:
2947:
2924:
2920:
2897:
2893:
2870:
2866:
2843:
2839:
2816:
2812:
2789:
2785:
2762:
2758:
2735:
2731:
2708:
2704:
2681:
2677:
2654:
2650:
2627:
2623:
2600:
2596:
2585:
2568:
2564:
2541:
2537:
2533:
2528:
2524:
2503:
2500:
2494:
2486:
2482:
2459:
2455:
2451:
2448:
2428:
2405:
2402:
2396:
2388:
2384:
2371:
2367:
2363:
2360:
2354:
2349:
2345:
2341:
2338:
2335:
2330:
2326:
2318:
2314:
2309:
2295:
2272:
2266:
2263:
2260:
2254:
2248:
2245:
2242:
2233:
2230:
2227:
2221:
2218:
2210:
2206:
2183:
2179:
2155:
2149:
2143:
2137:
2128:
2125:
2122:
2111:
2107:
2098:
2082:
2076:
2050:
2024:
2021:
2018:
1992:
1985:
1978:
1952:
1949:
1946:
1934:
1918:
1914:
1891:
1887:
1878:
1874:
1858:
1836:
1833:
1830:
1826:
1817:
1813:
1809:
1804:
1800:
1774:
1771:
1768:
1764:
1760:
1757:
1744:
1736:
1733:
1730:
1726:
1719:
1714:
1711:
1708:
1704:
1691:
1686:
1682:
1674:
1670:
1666:
1661:
1657:
1643:
1623:
1620:
1615:
1611:
1607:
1602:
1599:
1596:
1592:
1584:that satisfy
1583:
1565:
1561:
1538:
1534:
1511:
1508:
1505:
1501:
1492:
1488:
1484:
1479:
1475:
1466:
1450:
1425:
1421:
1417:
1412:
1408:
1397:
1396:
1395:chain complex
1391:
1387:
1371:
1348:
1340:
1336:
1321:
1319:
1301:
1297:
1289:and 2-sphere
1274:
1270:
1261:
1257:
1252:
1236:
1232:
1209:
1205:
1182:
1178:
1169:
1168:topologically
1159:
1157:
1153:
1149:
1145:
1140:
1138:
1134:
1130:
1126:
1122:
1118:
1113:
1111:
1107:
1103:
1093:
1089:
1087:
1083:
1079:
1075:
1071:
1067:
1062:
1058:
1054:
1049:
1047:
1043:
1039:
1035:
1031:
1026:
1016:
1014:
1011:. Tools from
1010:
992:
988:
979:
963:
941:
937:
916:
909:
905:
887:
883:
857:
854:
851:
847:
843:
840:
827:
819:
816:
813:
809:
802:
797:
794:
791:
787:
774:
769:
765:
756:
738:
734:
726:
710:
687:
684:
678:
670:
666:
653:
649:
645:
642:
636:
631:
627:
623:
620:
617:
612:
608:
599:
581:
577:
569:
553:
545:
527:
523:
518:
512:
508:
504:
499:
495:
487:
469:
465:
444:
421:
418:
413:
410:
407:
403:
399:
394:
390:
385:
382:
373:
370:
367:
363:
348:
345:
342:
338:
328:
324:
309:
305:
295:
292:
289:
285:
270:
267:
264:
260:
253:
250:
245:
241:
233:
232:
231:
229:
225:
224:boundary maps
207:
203:
195:
191:
173:
169:
161:
140:
136:
132:
127:
123:
111:
107:
106:chain complex
97:
95:
91:
87:
83:
79:
75:
71:
67:
63:
59:
55:
54:
53:chain complex
48:
44:
40:
33:
19:
14886:Publications
14751:Chern number
14741:Betti number
14624: /
14615:Key concepts
14579:
14563:Differential
14480:
14458:, Elsevier,
14455:
14410:
14380:
14358:
14345:
14342:Richeson, D.
14318:
14314:
14294:
14273:
14238:
14211:
14182:
14165:
14160:
14153:Teicher 1999
14143:
14133:
14131:For example
14127:
14115:
14103:
14091:
14080:
14068:
14048:
14026:
14014:
13981:
13977:
13967:
13932:
13928:
13877:
13873:
13860:
13848:. Retrieved
13839:
13827:. Retrieved
13823:the original
13813:
13806:Hatcher 2002
13801:
13794:Spanier 1966
13789:
13782:Hatcher 2002
13777:
13770:Hatcher 2002
13765:
13758:Hatcher 2002
13753:
13738:
13728:
13721:Hatcher 2002
13698:
13688:
13664:Spanier 1966
13659:
13584:Betti number
13551:
13542:
13525:
13505:Emmy Noether
13503:
13496:
13461:
13457:
13422:
13412:
13399:
13389:
13364:
13363:cross-caps (
13357:
13350:
13333:
13322:
13310:
13309:Sphere with
13294:
13282:
13268:
13264:
13210:Klein bottle
12987:(1-manifold)
12946:
12938:
12930:
12888:
12879:
12875:
12865:
12860:
12856:
12852:
12846:
12841:
12837:
12833:
12829:
12825:
12821:
12817:
12813:
12809:
12801:
12797:
12793:
12789:
12785:
12781:
12777:
12773:
12769:
12767:
12762:
12758:
12754:
12750:
12746:
12742:
12739:Klein bottle
12734:
12730:
12728:
12723:
12718:
12714:
12710:
12706:
12702:
12666:
12662:MΓΆbius strip
12657:
12653:
12649:
12645:
12641:
12637:
12630:Klein bottle
12601:
12589:
12584:
12580:
12576:
12572:
12570:
12565:
12561:
12557:
12553:
12549:
12545:
12513:
12507:has trivial
12477:
12469:
12467:great circle
12462:
12461:, the curve
12430:
12256:
12242:
12189:CAPD::Redhom
12162:
12134:
12114:Morse theory
12099:
12088:
12078:embedded in
12065:
11947:
11939:
11897:
11886:
11882:
11874:
11873:is open and
11754:
11674:
11670:
11660:
11652:Applications
11567:
11353:
11300:
11295:
11273:
11183:
11098:
11091:
11030:
11020:
11016:
11012:
11008:
11004:
11000:
10994:
10989:
10978:
10977:), then the
10974:
10970:
10940:
10938:
10926:
10922:
10918:
10669:
10559:
10548:
10544:
10540:
10493:
10488:
10484:
10480:
10476:
10472:
10441:
10206:
10202:
10198:
10195:Tor functors
10184:
10167:
10162:
10154:
10143:
10139:
10135:
10128:
10126:
10102:
10098:
10094:
10090:
10088:
9969:
9962:
9958:
9954:
9811:
9783:
9601:
9509:
9502:
9286:
9258:
9254:
9243:
9239:
9230:
9224:
9209:
8864:figure eight
8815:
8313:
8303:
8301:
8208:
8204:
8201:coboundaries
8125:
8058:
8054:
8018:
8013:
8009:
8002:
7996:
7994:
7985:
7981:
7973:
7966:
7962:
7958:
7954:
7949:
7945:
7938:
7932:
7930:
7855:
7850:
7682:
7607:
7498:
7282:
7278:
7272:
7267:
7263:
7259:
7255:
7249:
7244:
7240:
7236:
7229:
7224:
7220:
7219:
7217:
6857:
6811:
6809:
6804:
6739:
6622:denotes the
6502:
6431:
6343:
6339:
6304:
6092:
5964:
5960:
5956:
5947:
5945:
5934:
5924:
5828:
5708:cyclic group
5674:
5669:
5631:
5620:
5492:
5434:
5403:
5399:
5397:
5203:
5103:
5069:
4984:
4744:
4742:
4605:
4542:. The group
4350:
4255:
4211:
4207:
4046:
4042:
4037:is a set of
4034:
4029:
3903:
3700:th homology
3593:
3333:
3196:homeomorphic
3067:is given by
3022:
2586:
2312:
2310:
1935:
1668:
1662:
1658:
1393:
1389:
1385:
1327:
1317:
1255:
1253:
1165:
1141:
1129:homeomorphic
1114:
1101:
1099:
1090:
1050:
1024:
1022:
724:
723:th group of
567:
566:th group of
436:
109:
103:
77:
69:
65:
61:
50:
42:
36:
14849:Wikiversity
14766:Key results
14407:Teicher, M.
14120:Hilton 1988
14110:, p. 4
14108:Weibel 1999
14031:Weibel 1999
13589:Cycle space
13554:"identical"
12699:Boy surface
12122:KAM theorem
12072:point cloud
11759:open subset
11184:(using the
8910:of rank 2,
8630:. That is,
7218:called the
6803:are called
6738:are called
6095:. That is,
2421:, for some
1851:, for some
1078:resolutions
41:, the term
39:mathematics
14695:CW complex
14636:Continuity
14626:Closed set
14585:cohomology
13651:References
13356:holes and
13314:cross-caps
13287:Orientable
13245:Orientable
13108:Orientable
13052:Orientable
12996:Orientable
12228:Cohomology
12104:theory in
11946:Euclidean
11027:Properties
11003:manner on
10997:cohomology
10845:such that
10639:K-homology
10562:, so that
10238:surjective
10153:maps from
10151:continuous
9604:formal sum
8908:free group
7795:The extra
7651:of points
7245:homologous
6974:therefore
6740:boundaries
4933:The torus
4216:components
4134:where the
3508:, so such
3198:to a disk
1875:minus its
1390:boundaries
725:boundaries
703:, and the
542:of cycles
86:cohomology
66:homologies
14874:geometric
14869:algebraic
14720:Cobordism
14656:Hausdorff
14651:connected
14568:Geometric
14558:Continuum
14548:Algebraic
14441:223099225
14006:122763537
13984:: 1β155.
13945:CiteSeerX
13460:tori and
12902:Euler no.
12898:Manifold
12741:diagram,
12005:⊆
11970:⊆
11911:≥
11833:injective
11807:→
11688:∈
11606:−
11595:→
11553:⋯
11550:→
11533:−
11522:→
11505:−
11494:→
11477:−
11466:→
11449:−
11438:→
11416:→
11394:→
11372:→
11369:⋯
11336:→
11330:→
11324:→
11318:→
11282:χ
11215:−
11209:∑
11203:χ
11125:−
11119:∑
11113:χ
11073:−
11062:→
10954:→
10895:∘
10869:∘
10861:−
10823:→
10775:−
10764:→
10716:−
10705:→
10353:
10347:→
10271:→
10159:simplices
10062:σ
10049:∂
10023:∈
10014:σ
10009:∑
9987:∂
9905:σ
9879:∈
9870:σ
9865:∑
9828:∈
9752:σ
9746:…
9725:σ
9713:−
9704:σ
9698:…
9683:σ
9662:−
9639:∑
9629:σ
9617:∂
9575:σ
9569:…
9554:σ
9539:σ
9530:σ
9483:∂
9433:σ
9404:σ
9398:⋯
9383:σ
9368:σ
9336:σ
9330:…
9315:σ
9300:σ
9273:σ
9182:∈
8997:×
8989:≅
8945:∗
8937:≅
8919:π
8879:π
8782:π
8710:π
8670:→
8652:π
8643:∗
8543:→
8525:π
8516:∗
8452:π
8326:π
8156:−
8072:
7834:⟶
7828:∅
7780:≠
7703:~
7660:σ
7630:σ
7616:∑
7577:∑
7559:σ
7539:∑
7530:ϵ
7507:ϵ
7479:⟶
7468:ϵ
7463:⟶
7439:∂
7433:⟶
7409:∂
7403:⟶
7397:⋯
7387:−
7380:∂
7374:⟶
7363:−
7344:∂
7338:⟶
7308:∂
7302:⟶
7296:⋯
7139:∂
7110:∂
7103:
7043:∂
7036:
6994:∂
6953:∂
6946:
6940:⊆
6922:∂
6881:∂
6874:
6835:∂
6828:
6782:∂
6775:
6711:∂
6644:∂
6637:
6595:∂
6554:∂
6547:
6541:⊆
6523:∂
6481:−
6409:−
6371:∂
6367:∘
6358:∂
6323:≡
6279:∂
6273:⟶
6249:∂
6243:⟶
6219:∂
6213:⟶
6207:⋯
6197:−
6190:∂
6184:⟶
6173:−
6154:∂
6148:⟶
6118:∂
6112:⟶
6106:⋯
6071:−
6060:→
6038:∂
6014:…
5807:otherwise
5648:×
5599:otherwise
5577:≤
5571:≤
5376:otherwise
5341:×
5243:×
5182:otherwise
4957:×
4862:otherwise
4721:otherwise
4601:generator
4470:otherwise
4122:…
3902:, called
3826:-cycles (
3643:∙
3630:∙
3557:embedding
3329:cobordism
3254:manifolds
2452:∈
2364:∈
2342:
2267:⋅
2261:−
2249:⋅
2222:⋅
2171:. (Since
2150:−
1986:⨿
1834:−
1823:→
1761:∈
1608:∘
1600:−
1509:−
1498:→
1426:∙
1413:∙
1125:simplices
1059:from the
844:∈
646:∈
624:
400:∘
383:⋯
371:−
358:⟶
346:−
319:⟶
280:⟶
257:⟶
254:⋯
246:∙
230:is zero:
141:∙
128:∙
14901:Category
14839:Wikibook
14817:Category
14705:Manifold
14673:Homotopy
14631:Interior
14622:Open set
14580:Homology
14529:Topology
14357:(1966),
14344:(2008),
14210:(1956).
14171:Archived
13904:16865053
13850:16 March
13829:16 March
13745:Archived
13705:Archived
13562:See also
12707:A′
12267:Surfaces
12220:homotopy
12192:Archived
12159:Software
11937:from an
11881:between
10986:functors
10917:for all
10672:category
10547:-module
10131:for any
9961:and the
9263:vertices
8393:-sphere
8126:cocycles
7965:), with
6575:, where
5670:For the
3258:Embedded
2858:to that
2723:to that
1877:interior
1669:boundary
1636:for all
1260:embedded
1061:category
757:subgroup
600:subgroup
222:(called
43:homology
14864:general
14666:uniform
14646:compact
14597:Digital
14335:2689545
14257:1361982
13986:Bibcode
13937:Bibcode
13882:Bibcode
13740:YouTube
13700:YouTube
13378:
13319:
13279:
13271:is the
13215:
13159:
13103:
13047:
12991:
12924:Symbol
12806:torsion
12765:) = 0.
12717:is. If
12628:is the
12360:bottle
12249:Riemann
12239:Origins
12200:Perseus
12106:physics
11942:-sphere
11838:, then
11015:) form
10934:functor
9602:to the
9454:is the
9289:-tuple
9249:is the
8862:is the
8816:For an
8001:groups
7948:) of a
7937:groups
5925:torsion
5706:is the
5210:product
4538:is the
4210:with a
3147:, then
2642:, and
1873:closure
1057:functor
906:over a
904:modules
60:called
14859:Topics
14661:metric
14536:Fields
14462:
14439:
14429:
14395:
14365:
14333:
14302:
14281:
14255:
14245:
14230:529171
14228:
14218:
14056:
14004:
13947:
13902:
13446:Notes
13431:
13427:
13420:
13416:
13397:
13393:
13386:
13382:
13372:
13368:
13361:
13354:
13042:Sphere
12985:Circle
12433:sphere
12165:Linbox
12130:braids
11950:-space
11933:: any
11831:is an
11757:is an
11301:Every
10921:. The
10236:and a
10107:matrix
9926:where
9285:as an
9229:: the
8499:, the
7988:. See
7976:) the
6805:cycles
6667:kernel
4608:sphere
4490:where
4383:circle
4353:sphere
3906:, are
2966:has a
2556:, and
2317:kernel
1386:cycles
1256:cycles
1040:, and
956:to be
902:to be
598:kernel
568:cycles
544:modulo
192:) and
190:chains
14641:Space
14452:(PDF)
14331:JSTOR
14002:S2CID
13925:(PDF)
13900:S2CID
13870:(PDF)
13552:homos
13550:α½ΞΌΟΟ
13548:Greek
13534:Notes
13302:None
13281:2 β 2
13273:genus
13257:None
13120:None
13098:Torus
13064:None
12927:Name
12568:can.
12516:torus
12261:Betti
12253:genus
12212:Kenzo
12185:Chomp
12181:Maple
12167:is a
11944:into
11877:is a
11753:: If
11710:with
11669:: If
10551:as a
10161:into
9501:from
9235:of a
7252:exact
7021:is a
6624:image
5621:(see
5206:torus
4381:is a
4204:holes
3702:group
3127:onto
2313:cycle
1673:image
1392:of a
1074:exact
755:image
88:of a
14460:ISBN
14437:OCLC
14427:ISBN
14393:ISBN
14363:ISBN
14300:ISBN
14279:ISBN
14253:OCLC
14243:ISBN
14226:OCLC
14216:ISBN
14054:ISBN
13852:2014
13831:2014
13511:and
13370:>
13321:2 β
12878:(2)
12800:+ (β
12761:+ (β
12733:and
12705:and
12667:The
12640:and
12575:and
12560:and
12514:The
12255:and
12224:Gmsh
12206:and
12198:and
12179:and
11994:and
11929:The
11892:The
11885:and
11790:and
11665:The
11644:and
11186:rank
11031:If (
10475:and
9401:<
9395:<
9380:<
8585:>
8314:The
8128:and
7995:The
7931:The
7769:for
7273:The
6665:its
6338:for
5939:and
5625:and
5204:The
4987:ball
3025:cone
2095:The
1667:, a
1580:are
1388:and
437:The
228:maps
14385:doi
14323:doi
14151:in
13994:doi
13955:doi
13890:doi
13878:259
13456:of
13336:β 1
12874:",
12840:= 2
12814:can
12552:or
12218:of
12177:Gap
12169:C++
12100:In
12089:In
12066:In
11761:of
10350:ker
10185:In
9512:nβ1
9508:to
8420:to
8199:of
8124:of
8069:ker
7519:is
7100:ker
7033:ker
7025:of
6943:ker
6871:ker
6825:ker
6772:ker
6634:ker
6544:ker
5398:If
4206:in
4041:of
3806:of
3559:of
3481:in
3434:in
3194:is
2339:ker
2308:.)
1906:is
1084:or
621:ker
158:of
68:or
37:In
14903::
14435:,
14425:,
14391:,
14329:,
14319:60
14317:,
14267:;
14251:.
14224:.
14206:;
14038:^
14000:.
13992:.
13982:15
13980:.
13976:.
13953:.
13943:.
13933:35
13931:.
13927:.
13912:^
13898:.
13888:.
13876:.
13872:.
13743:.
13737:.
13713:^
13703:.
13697:.
13671:^
13439:2
13411:(2
13388:(2
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13232:2
13176:2
13008:β
12906:Ο
12904:,
12836:+
12820:+
12796:=
12792:+
12788:=
12726:.
12587:.
12548:,
12187:,
12183:.
12139:,
12086:.
11648:.
10491:.
10113:.
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9202:.
8870:,
7853:.
7783:0.
7247:.
7097::=
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6807:.
6346:,
5943:.
5927:.
5490:,
4258:.
3020:.
2615:,
2355::=
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1933:.
1720::=
1692::=
1656:.
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1088:.
1036:,
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803::=
775::=
637::=
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422:0.
96:.
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14514:t
14507:v
14469:.
14421:/
14417:/
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14350:.
14325::
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14232:.
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13833:.
13477:2
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13005:β
13002:1
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12993:0
12968:1
12964:S
12950:2
12947:b
12942:1
12939:b
12934:0
12931:b
12880:1
12861:n
12857:n
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12842:b
12838:b
12834:b
12830:b
12826:b
12822:b
12818:b
12810:b
12802:a
12798:a
12794:a
12790:a
12786:a
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12770:a
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12724:c
12719:b
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12703:A
12683:2
12679:P
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12650:b
12646:c
12642:b
12638:a
12614:2
12610:K
12585:b
12581:a
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12562:b
12558:a
12554:c
12550:b
12546:a
12530:2
12526:T
12493:2
12489:S
12478:c
12470:a
12463:b
12447:2
12443:S
12411:2
12407:P
12373:2
12369:K
12333:2
12329:T
12295:2
12291:S
12257:n
12046:.
12043:n
12040:=
12037:m
12015:n
12010:R
12002:V
11980:m
11975:R
11967:U
11948:n
11940:n
11914:1
11908:k
11898:k
11889:.
11887:V
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11875:f
11861:)
11858:U
11855:(
11852:f
11849:=
11846:V
11817:n
11812:R
11804:U
11801::
11798:f
11776:n
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11755:U
11736:.
11733:a
11730:=
11727:)
11724:a
11721:(
11718:f
11696:n
11692:B
11685:a
11675:B
11671:f
11620:)
11617:A
11614:(
11609:1
11603:n
11599:H
11592:)
11589:C
11586:(
11581:n
11577:H
11547:)
11544:A
11541:(
11536:2
11530:n
11526:H
11519:)
11516:C
11513:(
11508:1
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11488:B
11485:(
11480:1
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11470:H
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11460:A
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11432:C
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11410:B
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11388:A
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11380:n
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11315:0
11296:X
11259:)
11254:n
11250:H
11246:(
11242:k
11239:n
11236:a
11233:r
11226:n
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11218:1
11212:(
11206:=
11169:)
11164:n
11160:A
11156:(
11152:k
11149:n
11146:a
11143:r
11136:n
11132:)
11128:1
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11116:=
11094:n
11092:A
11076:1
11070:n
11066:A
11057:n
11053:A
11049::
11044:n
11040:d
11021:X
11013:H
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10981:n
10979:H
10975:Y
10971:X
10957:Y
10951:X
10941:X
10929:n
10927:H
10923:n
10919:n
10903:n
10899:f
10890:n
10886:e
10882:=
10877:n
10873:d
10864:1
10858:n
10854:f
10831:n
10827:B
10818:n
10814:A
10810::
10805:n
10801:f
10778:1
10772:n
10768:B
10759:n
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10751::
10746:n
10742:e
10719:1
10713:n
10709:A
10700:n
10696:A
10692::
10687:n
10683:d
10587:.
10584:M
10580:/
10576:E
10573:=
10570:G
10560:G
10549:M
10545:G
10541:E
10524:)
10521:M
10518:,
10515:G
10512:(
10507:2
10503:H
10489:X
10485:F
10481:n
10477:X
10473:F
10457:n
10453:H
10442:F
10426:n
10422:p
10399:n
10395:F
10374:.
10370:)
10365:1
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10357:(
10342:2
10338:F
10334::
10329:2
10325:p
10302:2
10298:F
10277:.
10274:X
10266:1
10262:F
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10253:1
10249:p
10222:1
10218:F
10207:X
10203:X
10199:F
10168:n
10163:X
10155:n
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10144:C
10140:X
10136:X
10103:n
10099:X
10095:X
10091:n
10074:.
10071:)
10066:i
10058:(
10053:n
10043:i
10039:m
10031:n
10027:X
10018:i
10005:=
10002:)
9999:c
9996:(
9991:n
9972:n
9970:C
9965:i
9963:m
9959:X
9955:n
9939:n
9935:X
9914:,
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9899:i
9895:m
9887:n
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9858:c
9836:n
9832:C
9825:c
9814:n
9812:C
9795:=
9792:n
9769:,
9765:)
9761:]
9758:n
9755:[
9749:,
9743:,
9740:]
9737:1
9734:+
9731:i
9728:[
9722:,
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9533:=
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9312:,
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8101:n
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7569:)
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7052:)
7047:n
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3724:X
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3382:X
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3313:2
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3273:R
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3211:2
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2624:S
2601:1
2597:S
2569:2
2565:T
2542:2
2538:S
2534:,
2529:1
2525:S
2504:0
2501:=
2498:)
2495:c
2492:(
2487:n
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2460:n
2456:C
2449:c
2429:n
2409:}
2406:0
2403:=
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2397:c
2394:(
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2379:|
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2358:{
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2255:1
2252:{
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2243:=
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2225:[
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2153:{
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2144:1
2141:{
2138:=
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2132:]
2129:1
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2123:0
2120:[
2117:(
2112:1
2108:d
2083:.
2080:}
2077:0
2074:{
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2051:1
2048:{
2028:]
2025:1
2022:,
2019:0
2016:[
1996:}
1993:1
1990:{
1982:}
1979:0
1976:{
1956:]
1953:1
1950:,
1947:0
1944:[
1919:1
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1892:2
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1818:n
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1810::
1805:n
1801:d
1780:}
1775:1
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1758:c
1753:|
1748:)
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1742:(
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1723:{
1715:1
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1709:n
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1644:n
1624:0
1621:=
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1612:d
1603:1
1597:n
1593:d
1566:n
1562:d
1539:n
1535:C
1512:1
1506:n
1502:C
1493:n
1489:C
1485::
1480:n
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1451:X
1431:)
1422:d
1418:,
1409:C
1405:(
1372:X
1352:)
1349:X
1346:(
1341:n
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1302:2
1298:S
1275:2
1271:T
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993:n
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976:-
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806:{
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735:B
711:n
691:}
688:0
685:=
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679:c
676:(
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640:{
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524:B
519:/
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419:=
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374:1
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251::
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208:n
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174:n
170:C
146:)
137:d
133:,
124:C
120:(
34:.
20:)
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