10823:
10606:
10844:
10812:
10881:
10854:
10834:
3222:
1723:
7160:
7924:
observed that it is fruitful to omit the dimension axiom completely: this gives the notion of a generalized homology theory or a generalized cohomology theory, defined below. There are generalized cohomology theories such as K-theory or complex cobordism that give rich information about a topological
7881:, and any two constructions that share those properties will agree at least on all CW complexes. There are versions of the axioms for a homology theory as well as for a cohomology theory. Some theories can be viewed as tools for computing singular cohomology for special topological spaces, such as
2722:
constructed an integral cohomology class of degree 7 on a smooth 14-manifold that is not the class of any smooth submanifold. On the other hand, he showed that every integral cohomology class of positive degree on a smooth manifold has a positive multiple that is the class of a smooth submanifold.
9284:
induces an isomorphism on homology or cohomology. (That is true for singular homology or singular cohomology, but not for sheaf cohomology, for example.) Since every space admits a weak homotopy equivalence from a CW complex, this axiom reduces homology or cohomology theories on all spaces to the
9273:) that induce the zero map between homology theories on CW-pairs. Likewise, the functor from the stable homotopy category to generalized cohomology theories on CW-pairs is not an equivalence. It is the stable homotopy category, not these other categories, that has good properties such as being
8376:
737:
7877:). (Here sheaf cohomology is considered only with coefficients in a constant sheaf.) These theories give different answers for some spaces, but there is a large class of spaces on which they all agree. This is most easily understood axiomatically: there is a list of properties known as the
73:, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout
9268:
A subtle point is that the functor from the stable homotopy category (the homotopy category of spectra) to generalized homology theories on CW-pairs is not an equivalence, although it gives a bijection on isomorphism classes; there are nonzero maps in the stable homotopy category (called
8955:
7533:
1546:
337:
7292:
7700:
1331:
2750:. The cup product corresponds to the product of differential forms. This interpretation has the advantage that the product on differential forms is graded-commutative, whereas the product on singular cochains is only graded-commutative up to
6672:
In a broad sense of the word, "cohomology" is often used for the right derived functors of a left exact functor on an abelian category, while "homology" is used for the left derived functors of a right exact functor. For example, for a ring
6993:
8217:
601:
8790:
1528:
9264:
says that every generalized homology theory comes from a spectrum, and likewise every generalized cohomology theory comes from a spectrum. This generalizes the representability of ordinary cohomology by
Eilenberg–MacLane spaces.
6907:, more powerful tools are required because the classical definitions of homology/cohomology break down. This is because varieties over finite fields will only be a finite set of points. Grothendieck came up with the idea for a
8669:
9246:
7847:
3637:
7451:
4436:
2306:
216:
5972:
58:. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are
9086:
8509:
4594:
5733:
6191:
2059:
4240:
5849:
6356:
5037:
5436:
2162:
483:
2723:
Also, every integral cohomology class on a manifold can be represented by a "pseudomanifold", that is, a simplicial complex that is a manifold outside a closed subset of codimension at least 2.
591:
1188:
2429:
1718:{\displaystyle 0\to \operatorname {Ext} _{\mathbb {Z} }^{1}(\operatorname {H} _{i-1}(X,\mathbb {Z} ),A)\to H^{i}(X,A)\to \operatorname {Hom} _{\mathbb {Z} }(H_{i}(X,\mathbb {Z} ),A)\to 0.}
5594:
1134:
1059:
7171:
5532:
5195:
7384:
2996:
1166:
is an important computational tool in cohomology, as in homology. Note that the boundary homomorphism increases (rather than decreases) degree in cohomology. That is, if a space
7544:
5124:
9384:
9337:
8165:
8109:
6878:
2644:
7326:
4889:
2343:
1401:
9280:
If one prefers homology or cohomology theories to be defined on all topological spaces rather than on CW complexes, one standard approach is to include the axiom that every
2804:
510:
3129:
2774:
1385:
9474:
6982:
5472:
4942:
3433:
3338:
9518:
7440:
7155:{\displaystyle H^{k}(X;\mathbb {Q} _{\ell }):=\varprojlim _{n\in \mathbb {N} }H_{et}^{k}(X;\mathbb {Z} /(\ell ^{n}))\otimes _{\mathbb {Z} _{\ell }}\mathbb {Q} _{\ell }.}
5382:
5312:
5259:
1878:
1821:
1772:
9686:
9611:
9572:
9429:
3035:
835:
9754:
9647:
2908:
2860:
984:
8580:
3171:
3081:
7414:
4812:
7346:
6954:
6847:, help give computations of cohomology of these types of varieties (with the addition of more refined information). In the simplest case the cohomology of a smooth
5226:
5151:
3381:
3260:
9157:
6205:
Although cohomology is fundamental to modern algebraic topology, its importance was not seen for some 40 years after the development of homology. The concept of
7882:
7723:
6933:
6905:
6837:
2446:
Here are some of the geometric interpretations of the cup product. In what follows, manifolds are understood to be without boundary, unless stated otherwise. A
5076:
3528:
4319:
2185:
9797:
9001:
8424:
8371:{\displaystyle \cdots \to h_{i}(A){\overset {f_{*}}{\to }}h_{i}(X){\overset {g_{*}}{\to }}h_{i}(X,A){\overset {\partial }{\to }}h_{i-1}(A)\to \cdots .}
732:{\displaystyle \cdots \leftarrow C_{i+1}^{*}{\stackrel {d_{i}}{\leftarrow }}\ C_{i}^{*}{\stackrel {d_{i-1}}{\leftarrow }}C_{i-1}^{*}\leftarrow \cdots }
10884:
6113:
7731:
1957:
8950:{\displaystyle \cdots \to h^{i}(X,A){\overset {g_{*}}{\to }}h^{i}(X){\overset {f_{*}}{\to }}h^{i}(A){\overset {d}{\to }}h^{i+1}(X,A)\to \cdots .}
4451:
Another interpretation of
Poincaré duality is that the cohomology ring of a closed oriented manifold is self-dual in a strong sense. Namely, let
17:
7528:{\displaystyle {\begin{matrix}E&\longrightarrow &Bl_{Z}(X)\\\downarrow &&\downarrow \\Z&\longrightarrow &X\end{matrix}}}
4157:
8574:
7390:
which behave similarly to singular cohomology. There is a conjectured theory of motives which underlie all of the Weil cohomology theories.
9782:
Cohomology theories in a broader sense (invariants of other algebraic or geometric structures, rather than of topological spaces) include:
5879:
6459:, and realized that Poincaré duality can be stated in terms of the cap product. Their theory was still limited to finite cell complexes.
6070:
to make them intersect transversely and then taking the class of their intersection, which is a compact oriented submanifold of dimension
6915:
to define the cohomology theory for varieties over a finite field. Using the étale topology for a variety over a field of characteristic
6213:
used in his proof of his
Poincaré duality theorem, contained the beginning of the idea of cohomology, but this was not seen until later.
2087:
4489:
7925:
space, not directly accessible from singular cohomology. (In this context, singular cohomology is often called "ordinary cohomology".)
5628:
518:
7386:
whenever the variety is smooth over both fields. In addition to these cohomology theories there are other cohomology theories called
82:
9260:
determines both a generalized homology theory and a generalized cohomology theory. A fundamental result by Brown, Whitehead, and
5756:
332:{\displaystyle \cdots \to C_{i+1}{\stackrel {\partial _{i+1}}{\to }}C_{i}{\stackrel {\partial _{i}}{\to }}\ C_{i-1}\to \cdots }
6512:
is a rich generalization of singular cohomology, allowing more general "coefficients" than simply an abelian group. For every
6266:
10243:
347:
is the homology of this chain complex (the kernel of one homomorphism modulo the image of the previous one). In more detail,
9151:
4954:
2810:
to make the product graded-commutative on the nose. The failure of graded-commutativity at the cochain level leads to the
8679:
The axioms for a generalized cohomology theory are obtained by reversing the arrows, roughly speaking. In more detail, a
5387:
417:
10518:
9915:
7443:
6661:(but usually now just "cohomology"). From that point of view, sheaf cohomology becomes a sequence of functors from the
2565:
10872:
10867:
10461:
10435:
10405:
10348:
10313:
10280:
10210:
6653:
That definition suggests various generalizations. For example, one can define the cohomology of a topological space
3691:/2 coefficients works for arbitrary manifolds. With integer coefficients, the answer is a bit more complicated. The
2351:
9520:
and so on. Complex cobordism has turned out to be especially powerful in homotopy theory. It is closely related to
7870:
6492:
86:
5541:
10862:
9765:
7287:{\displaystyle X=\operatorname {Proj} \left({\frac {\mathbb {Z} \left}{\left(f_{1},\ldots ,f_{k}\right)}}\right)}
9939:
9792:
3225:
The first cohomology group of the 2-dimensional torus has a basis given by the classes of the two circles shown.
1068:
993:
85:
theory, is more natural than homology in many applications. At a basic level, this has to do with functions and
3459:= in degree 2. (Implicitly, orientations of the torus and of the two circles have been fixed here.) Note that
2714:
is a noncompact manifold, then a closed submanifold (not necessarily compact) determines a cohomology class on
10764:
10375:
9691:
7878:
6470:
6392:
5485:
2718:. In both cases, the cup product can again be described in terms of intersections of submanifolds. Note that
1533:
6730:) can be viewed as a "cohomology theory" in each variable, the right derived functors of the Hom functor Hom
2754:. In fact, it is impossible to modify the definition of singular cochains with coefficients in the integers
10905:
7695:{\displaystyle \cdots \to H^{n}(X)\to H^{n}(Z)\oplus H^{n}(Bl_{Z}(X))\to H^{n}(E)\to H^{n+1}(X)\to \cdots }
5156:
2559:
8691:) from the category of CW-pairs to the category of abelian groups, together with a natural transformation
10397:
10370:
7860:
7351:
4766:
2937:
2686:
do not intersect transversely, this formula can still be used to compute the cup product , by perturbing
1326:{\displaystyle \cdots \to H^{i}(X)\to H^{i}(U)\oplus H^{i}(V)\to H^{i}(U\cap V)\to H^{i+1}(X)\to \cdots }
1163:
6466:
overcame the technical limitations, and gave the modern definition of singular homology and cohomology.
1936:
On the other hand, cohomology has a crucial structure that homology does not: for any topological space
10772:
10340:
10272:
6217:
5092:
3518:
63:
9345:
9298:
8114:
8058:
6854:
9812:
9709:
Many of these theories carry richer information than ordinary cohomology, but are harder to compute.
9392:
groups, based on studying a space by considering all maps from it to manifolds: unoriented cobordism
9281:
7300:
4837:
4750:
2313:
1925:
2779:
10571:
6817:
3743:
1900:
manifold (possibly with boundary), or a CW complex with finitely many cells in each dimension, and
950:
Some of the formal properties of cohomology are only minor variants of the properties of homology:
488:
6477:, they proved that the existing homology and cohomology theories did indeed satisfy their axioms.
3086:
2757:
2542:, called . In these terms, the cup product describes the intersection of submanifolds. Namely, if
1342:
10857:
10843:
10138:
9872:
9434:
7865:
There are various ways to define cohomology for topological spaces (such as singular cohomology,
6961:
6415:
5441:
4906:
3398:
3303:
513:
160:
10365:
9481:
7419:
6634:, but not necessarily right exact. Grothendieck defined sheaf cohomology groups to be the right
6429:
and
Alexander both introduced cohomology and tried to construct a cohomology product structure.
5345:
5275:
5231:
1841:
1784:
1735:
10792:
10713:
10590:
10578:
10551:
10511:
10388:
9897:
9882:
9832:
9817:
9652:
9577:
9538:
9395:
9340:
7973:
7949:
7387:
6591:
4711:
3001:
808:
210:
148:
structure. Because of this feature, cohomology is usually a stronger invariant than homology.
59:
10787:
9723:
9616:
5264:
There is a related description of the first cohomology with coefficients in any abelian group
2877:
2829:
957:
10634:
10561:
9867:
9274:
6908:
6695:) form a "homology theory" in each variable, the left derived functors of the tensor product
3642:
3134:
3044:
2346:
1137:
10195:
7396:
4782:
10782:
10734:
10708:
10556:
10482:
10445:
10415:
10358:
10323:
10290:
10253:
10220:
7331:
6939:
6813:
6585:
5204:
5129:
4742:
4664:
4244:
Alternatively, the external product can be defined in terms of the cup product. For spaces
3359:
3238:
2731:
1541:
936:
8:
10629:
9702:
9257:
7921:
6806:
6595:
6513:
6225:
3793:
3279:
2440:
2436:
1726:
1182:
10833:
9837:
4631:) have the same (finite) dimension. Likewise, the product on integral cohomology modulo
4446:
3981:) have square zero. On the other hand, odd-degree elements need not have square zero if
2934:
with an orientation on the normal bundle. Informally, one thinks of the resulting class
2494:
1523:{\displaystyle \cdots \to H^{i}(X,Y)\to H^{i}(X)\to H^{i}(Y)\to H^{i+1}(X,Y)\to \cdots }
10827:
10797:
10777:
10698:
10688:
10566:
10546:
10486:
9902:
9892:
9827:
9787:
7894:
7886:
7708:
6918:
6890:
6822:
6631:
6573:
6403:
2811:
2179:
2061:
defined by an explicit formula on singular cochains. The product of cohomology classes
355:
145:
39:
9807:
7866:
6437:
5049:
3503:
10822:
10815:
10681:
10639:
10504:
10490:
10431:
10401:
10344:
10334:
10309:
10276:
10239:
10206:
9877:
9857:
9822:
9476:
7890:
6936:
6426:
6411:
6410:. This (in rather special cases) provided an interpretation of Poincaré duality and
6407:
6221:
6090:
5988:
be an oriented manifold, not necessarily compact. Then a closed oriented codimension-
5475:
2747:
1388:
1336:
910:
595:
This has the effect of "reversing all the arrows" of the original complex, leaving a
340:
179:
106:
51:
10847:
10190:
6912:
6210:
3957:
On any topological space, graded-commutativity of the cohomology ring implies that 2
10595:
10541:
10470:
10297:
10260:
9887:
9862:
9293:
8381:
7953:
7874:
6985:
6662:
6603:
6581:
6577:
6504:
6463:
6245:
5327:
4754:
4632:
3341:
925:
78:
5596:
corresponding to the trivial double covering, the disjoint union of two copies of
10654:
10649:
10478:
10441:
10427:
10411:
10354:
10319:
10305:
10286:
10264:
10249:
10235:
10216:
9847:
9802:
9695:
6844:
6747:
Sheaf cohomology can be identified with a type of Ext group. Namely, for a sheaf
6658:
6635:
6444:
6433:
6388:
4600:
4456:
3275:
2675:
2170:
1905:
596:
202:, the cohomology ring tends to be computable in practice for spaces of interest.
55:
35:
10837:
8664:{\displaystyle \bigoplus _{\alpha }h_{i}(X_{\alpha },A_{\alpha })\to h_{i}(X,A)}
6473:
defining a homology or cohomology theory, discussed below. In their 1952 book,
10744:
10676:
9852:
9525:
6810:
6541:
6399:
5315:
2751:
199:
164:
9955:
7393:
Another useful computational tool is the blowup sequence. Given a codimension
6448:
5314:
is in one-to-one correspondence with the set of isomorphism classes of Galois
10899:
10754:
10664:
10644:
10330:
10227:
9761:
6488:
6372:
Alexander had by 1930 defined a first notion of a cochain, by thinking of an
4670:
3507:
3271:
3209:
in degree 0. By homotopy invariance, this is also the cohomology ring of any
2703:
2486:
2462:
1897:
47:
10202:
9241:{\displaystyle h^{i}(X,A)\to \prod _{\alpha }h^{i}(X_{\alpha },A_{\alpha })}
7842:{\displaystyle H^{n}(Bl_{Z}(X))\oplus H^{n}(Z)\cong H^{n}(X)\oplus H^{n}(E)}
1398:. They are related to the usual cohomology groups by a long exact sequence:
10739:
10659:
10605:
10384:
9521:
6884:
6848:
6840:
4023:
3210:
2694:
to make the intersection transverse. More generally, without assuming that
1945:
1779:
10452:
9760:. In the language of spectra, there are several more precise notions of a
4154:) can be defined as the pullback of the external product by the diagonal:
3703:
of degree 2 such that the whole cohomology is the direct sum of a copy of
2719:
10749:
9270:
9261:
6678:
6456:
6452:
6362:
5609:
4746:
4710:). Informally, the Euler class is the class of the zero set of a general
4690:
3632:{\displaystyle H^{*}(X\times Y,R)\cong H^{*}(X,R)\otimes _{R}H^{*}(Y,R).}
3440:
2739:
2525:
2165:
1950:
1171:
141:
31:
6594:
elegantly defined and characterized sheaf cohomology in the language of
6572:). Starting in the 1950s, sheaf cohomology has become a central part of
10693:
10624:
10474:
9535:, based on studying a space by considering all vector bundles over it:
6481:
4431:{\displaystyle u\times v=(f^{*}(u))(g^{*}(v))\in H^{i+j}(X\times Y,R).}
3669:
3197:
In what follows, cohomology is taken with coefficients in the integers
2082:
1835:
1775:
412:
9774:, where the product is commutative and associative in a strong sense.
2301:{\displaystyle uv=(-1)^{ij}vu,\qquad u\in H^{i}(X,R),v\in H^{j}(X,R).}
10718:
10304:, Graduate Texts in Mathematics, vol. 52, New York, Heidelberg:
9842:
9389:
6713:
5967:{\displaystyle H^{i}(X,R)\to \operatorname {Hom} _{R}(H_{i}(X,R),R),}
1885:
1537:
1148:
10157:
7972:
is a subcomplex) to the category of abelian groups, together with a
7900:
One of the
Eilenberg–Steenrod axioms for a cohomology theory is the
7297:
then there is an equality of dimensions for the Betti cohomology of
4822:
and whose other homotopy groups are zero. Such a space is called an
10703:
10671:
10620:
10527:
10114:
10078:
9698:, Morava E-theory, and other theories built from complex cobordism.
9532:
5126:(defined up to homotopy equivalence) can be taken to be the circle
3233:
1831:
74:
70:
9081:{\displaystyle h^{i}(X,B){\overset {f_{*}}{\to }}h^{i}(A,A\cap B)}
8759:) induces a long exact sequence in cohomology, via the inclusions
8504:{\displaystyle h_{i}(A,A\cap B){\overset {f_{*}}{\to }}h_{i}(X,B)}
6805:
There are numerous machines built for computing the cohomology of
4589:{\displaystyle H^{i}(X,F)\times H^{n-i}(X,F)\to H^{n}(X,F)\cong F}
10102:
10032:
10030:
7933:
5728:{\displaystyle \cap :H^{i}(X,R)\times H_{j}(X,R)\to H_{j-i}(X,R)}
1159:
induce the same homomorphism on cohomology (just as on homology).
944:
8181:) induces a long exact sequence in homology, via the inclusions
7725:
is smooth, then the connecting morphisms are all trivial, hence
6395:. This result can be stated more simply in terms of cohomology.
6422:
6216:
There were various precursors to cohomology. In the mid-1920s,
6186:{\displaystyle H^{i}(X,R){\overset {\cong }{\to }}H_{n-i}(X,R)}
4022:
The cup product on cohomology can be viewed as coming from the
3221:
10456:
10054:
10027:
10015:
10003:
9967:
6809:. The simplest case being the determination of cohomology for
4284:
for the two projections. Then the external product of classes
7852:
4745:
for vector bundles that take values in cohomology, including
3707:
spanned by the element 1 in degree 0 together with copies of
3298:
140:. The most important cohomology theories have a product, the
9991:
8745:: Homotopic maps induce the same homomorphism on cohomology.
6657:
with coefficients in any complex of sheaves, earlier called
6260:)-cycle. This leads to a multiplication of homology classes
5078:
denotes the set of homotopy classes of continuous maps from
2054:{\displaystyle H^{i}(X,R)\times H^{j}(X,R)\to H^{i+j}(X,R),}
81:. The terminology tends to hide the fact that cohomology, a
27:
Sequences of abelian groups attached with topological spaces
10496:
10457:"Quelques propriétés globales des variétés différentiables"
10147:, p. 117, 331, Theorem 9.27; Corollary 14.36; Remarks.
6880:
can be determined from the degree of the polynomial alone.
857:
can be identified with a function from the set of singular
8167:, then the induced homomorphisms on homology are the same.
6568:
a manifold or CW complex (though not for arbitrary spaces
6491:, building on work of Alexander and Kolmogorov, developed
4235:{\displaystyle uv=\Delta ^{*}(u\times v)\in H^{i+j}(X,R).}
190:; this puts strong restrictions on the possible maps from
6793:, and Ext is taken in the abelian category of sheaves on
6610:
to abelian groups. Start with the functor taking a sheaf
5844:{\displaystyle H^{*}(X,R)\times H_{*}(X,R)\to H_{*}(X,R)}
6789:
denotes the constant sheaf associated with the integers
6380:
as a function on small neighborhoods of the diagonal in
6351:{\displaystyle H_{i}(M)\times H_{j}(M)\to H_{i+j-n}(M),}
6193:
is defined by cap product with the fundamental class of
2742:
coefficients is isomorphic to the de Rham cohomology of
2662:, with an orientation determined by the orientations of
209:, the definition of singular cohomology starts with the
10168:
6200:
3083:
restricts to zero in the cohomology of the open subset
2450:
means a compact manifold (without boundary), whereas a
10066:
10042:
9339:
The corresponding homology theory is used more often:
9288:
Some examples of generalized cohomology theories are:
7456:
5153:. So the description above says that every element of
4826:. This space has the remarkable property that it is a
10158:"Are spectra really the same as cohomology theories?"
9927:
9726:
9655:
9619:
9580:
9541:
9484:
9437:
9398:
9348:
9301:
9160:
9004:
8793:
8583:
8427:
8220:
8117:
8061:
7734:
7711:
7547:
7538:
From this there is an associated long exact sequence
7454:
7422:
7399:
7354:
7334:
7303:
7174:
6996:
6964:
6942:
6921:
6893:
6857:
6825:
6269:
6116:
5882:
5759:
5631:
5544:
5488:
5444:
5390:
5348:
5278:
5234:
5207:
5159:
5132:
5095:
5052:
4957:
4909:
4840:
4785:
4718:. That interpretation can be made more explicit when
4492:
4322:
4160:
3531:
3401:
3362:
3306:
3241:
3137:
3089:
3047:
3004:
2940:
2880:
2832:
2782:
2760:
2568:
2354:
2316:
2188:
2090:
1960:
1844:
1787:
1738:
1549:
1404:
1345:
1191:
1071:
996:
960:
811:
604:
521:
491:
420:
219:
9979:
6602:
and think of sheaf cohomology as a functor from the
5032:{\displaystyle {\stackrel {\cong }{\to }}H^{j}(X,A)}
3498:-module in each degree. (No assumption is needed on
6580:, partly because of the importance of the sheaf of
6000:(not necessarily compact) determines an element of
5858:into a module over the singular cohomology ring of
5431:{\displaystyle \operatorname {Hom} (\pi _{1}(X),A)}
3788:The cohomology ring of the closed oriented surface
3181:could be replaced by any continuous deformation of
2157:{\displaystyle H^{*}(X,R)=\bigoplus _{i}H^{i}(X,R)}
478:{\displaystyle C_{i}^{*}:=\mathrm {Hom} (C_{i},A),}
159:is a powerful invariant in topology, associating a
10194:
10090:
9756:has the structure of a graded ring for each space
9748:
9680:
9641:
9605:
9566:
9512:
9468:
9423:
9378:
9331:
9240:
9080:
8949:
8663:
8503:
8370:
8159:
8103:
7841:
7717:
7694:
7527:
7434:
7408:
7378:
7340:
7320:
7286:
7154:
6976:
6948:
6927:
6899:
6872:
6831:
6350:
6244:-cycle with nonempty intersection will, if in the
6185:
5966:
5843:
5727:
5588:
5526:
5466:
5430:
5376:
5306:
5253:
5220:
5189:
5145:
5118:
5070:
5031:
4936:
4883:
4806:
4588:
4430:
4234:
3631:
3427:
3375:
3332:
3254:
3165:
3123:
3075:
3029:
2990:
2902:
2854:
2798:
2768:
2638:
2423:
2337:
2300:
2156:
2053:
1872:
1815:
1766:
1717:
1522:
1379:
1325:
1128:
1053:
978:
829:
731:
585:
504:
477:
331:
10259:
10126:
6391:related homology and differential forms, proving
6361:which (in retrospect) can be identified with the
5873:, the cap product gives the natural homomorphism
4722:is a smooth vector bundle over a smooth manifold
3839:of a point in degree 2. The product is given by:
3687:even and positive, because Poincaré duality with
1536:describes cohomology in terms of homology, using
586:{\displaystyle d_{i-1}:C_{i-1}^{*}\to C_{i}^{*}.}
10897:
3803:-module of the form: the element 1 in degree 0,
2862:can be thought of as represented by codimension-
358:on the set of continuous maps from the standard
2874:. For example, one way to define an element of
920:is sometimes not written. It is common to take
10197:History of Algebraic and Differential Topology
9952:, Proposition VIII.3.3 and Corollary VIII.3.4.
6228:of cycles on manifolds. On a closed oriented
2443:, then their cohomology rings are isomorphic.
2424:{\displaystyle f^{*}:H^{*}(Y,R)\to H^{*}(X,R)}
1140:from topological spaces to abelian groups (or
10512:
6618:to its abelian group of global sections over
5046:with the homotopy type of a CW complex. Here
2520:. As a result, a closed oriented submanifold
9777:
5589:{\displaystyle 0\in H^{1}(X,\mathbb {Z} /2)}
2698:has an orientation, a closed submanifold of
1136:on cohomology. This makes cohomology into a
9106:) is the disjoint union of a set of pairs (
8529:) is the disjoint union of a set of pairs (
6469:In 1945, Eilenberg and Steenrod stated the
6455:(making cohomology into a graded ring) and
4830:for cohomology: there is a natural element
4760:
3969:containing 1/2, all odd-degree elements of
3931:. By graded-commutativity, it follows that
3731:is the same together with an extra copy of
3205:The cohomology ring of a point is the ring
2822:Very informally, for any topological space
10880:
10853:
10519:
10505:
10424:Algebraic Topology — Homology and Homotopy
10296:
10120:
7853:Axioms and generalized cohomology theories
6800:
6598:. The essential point is to fix the space
3961:= 0 for all odd-degree cohomology classes
1129:{\displaystyle f^{*}:H^{i}(Y)\to H^{i}(X)}
1054:{\displaystyle f_{*}:H_{i}(X)\to H_{i}(Y)}
10189:
10108:
9688:(complex connective K-theory), and so on.
9531:Various different flavors of topological
7363:
7311:
7195:
7139:
7125:
7090:
7053:
7018:
6860:
5571:
5534:classifies the double covering spaces of
5509:
5180:
5103:
4944:. More precisely, pulling back the class
4726:, since then a general smooth section of
4658:
3352:generators in degree 1. For example, let
3290:. In terms of Poincaré duality as above,
2784:
2762:
1693:
1661:
1608:
1562:
8683:is a sequence of contravariant functors
6564:) coincide with singular cohomology for
3447:of the form: the element 1 in degree 0,
3220:
198:. Unlike more subtle invariants such as
10421:
10329:
10174:
10144:
10084:
10072:
10060:
10048:
10036:
10021:
10009:
9973:
9945:
9933:
5527:{\displaystyle H^{1}(X,\mathbb {Z} /2)}
4891:, and every cohomology class of degree
916:In what follows, the coefficient group
14:
10898:
10390:A Concise Course in Algebraic Topology
9285:corresponding theory on CW complexes.
3506:gives that the cohomology ring of the
3294:is the class of a point on the sphere.
3041:; this is justified in that the class
885:, respectively, while elements of ker(
384:-th boundary homomorphism. The groups
151:
89:in geometric situations: given spaces
10500:
5190:{\displaystyle H^{1}(X,\mathbb {Z} )}
2734:says that the singular cohomology of
10451:
10226:
9997:
9985:
9961:
9949:
6540:. In particular, in the case of the
6201:Brief history of singular cohomology
6110:). The Poincaré duality isomorphism
4818:-th homotopy group is isomorphic to
3482:be any topological spaces such that
2469:, not necessarily compact (although
2439:. It follows that if two spaces are
46:is a general term for a sequence of
10383:
10132:
10096:
7379:{\displaystyle X(\mathbb {F} _{q})}
7165:If we have a scheme of finite type
6911:and used sheaf cohomology over the
6498:
4607:. In particular, the vector spaces
4440:
2991:{\displaystyle f^{*}()\in H^{i}(X)}
24:
9964:, Propositions IV.8.12 and V.4.11.
9916:complex-oriented cohomology theory
8329:
6883:When considering varieties over a
4171:
3777:is the class of a linear subspace
1725:A related statement is that for a
1580:
493:
446:
443:
440:
291:
252:
163:with any topological space. Every
25:
10917:
10462:Commentarii Mathematici Helvetici
10269:Foundations of Algebraic Topology
8573:) induce an isomorphism from the
6548:associated with an abelian group
6475:Foundations of Algebraic Topology
5119:{\displaystyle K(\mathbb {Z} ,1)}
4741:There are several other types of
4684:determines a cohomology class on
2706:determines a cohomology class on
2535:determines a cohomology class in
931:; then the cohomology groups are
10879:
10852:
10842:
10832:
10821:
10811:
10810:
10604:
9379:{\displaystyle \pi _{*}^{S}(X).}
9332:{\displaystyle \pi _{S}^{*}(X).}
8160:{\displaystyle g:(X,A)\to (Y,B)}
8104:{\displaystyle f:(X,A)\to (Y,B)}
6873:{\displaystyle \mathbb {P} ^{n}}
6248:, have as their intersection a (
6062:) can be computed by perturbing
3993:, as one sees in the example of
2654:is a submanifold of codimension
2639:{\displaystyle =\in H^{i+j}(X),}
2550:are submanifolds of codimension
939:. A standard choice is the ring
50:, usually one associated with a
10150:
9150:) induce an isomorphism to the
7321:{\displaystyle X(\mathbb {C} )}
6887:, or a field of characteristic
5854:makes the singular homology of
4884:{\displaystyle H^{j}(K(A,j),A)}
4459:oriented manifold of dimension
4017:
3679:; this makes sense even though
3494:) is a finitely generated free
3474:be a commutative ring, and let
3213:space, such as Euclidean space
2338:{\displaystyle f\colon X\to Y,}
2229:
18:Generalized cohomology theories
10232:Lectures on Algebraic Topology
9743:
9737:
9675:
9669:
9636:
9630:
9600:
9594:
9561:
9555:
9504:
9498:
9460:
9454:
9418:
9412:
9370:
9364:
9323:
9317:
9235:
9209:
9186:
9183:
9171:
9075:
9057:
9032:
9027:
9015:
8938:
8935:
8923:
8899:
8894:
8888:
8863:
8858:
8852:
8827:
8822:
8810:
8797:
8658:
8646:
8633:
8630:
8604:
8498:
8486:
8461:
8456:
8438:
8359:
8356:
8350:
8326:
8321:
8309:
8284:
8279:
8273:
8248:
8243:
8237:
8224:
8154:
8142:
8139:
8136:
8124:
8098:
8086:
8083:
8080:
8068:
7836:
7830:
7814:
7808:
7792:
7786:
7770:
7767:
7761:
7745:
7686:
7683:
7677:
7658:
7655:
7649:
7636:
7633:
7630:
7624:
7608:
7592:
7586:
7573:
7570:
7564:
7551:
7513:
7501:
7495:
7488:
7482:
7464:
7373:
7358:
7315:
7307:
7115:
7112:
7099:
7080:
7028:
7007:
6342:
6336:
6311:
6308:
6302:
6286:
6280:
6180:
6168:
6144:
6139:
6127:
5958:
5949:
5937:
5924:
5908:
5905:
5893:
5838:
5826:
5813:
5810:
5798:
5782:
5770:
5722:
5710:
5691:
5688:
5676:
5660:
5648:
5603:
5583:
5561:
5521:
5499:
5461:
5455:
5425:
5416:
5410:
5397:
5371:
5359:
5301:
5289:
5238:
5197:is pulled back from the class
5184:
5170:
5113:
5099:
5065:
5053:
5026:
5014:
4992:
4985:
4982:
4970:
4958:
4931:
4919:
4913:
4878:
4869:
4857:
4851:
4801:
4789:
4577:
4565:
4552:
4549:
4537:
4515:
4503:
4422:
4404:
4382:
4379:
4373:
4360:
4357:
4354:
4348:
4335:
4226:
4214:
4192:
4180:
4122:). The cup product of classes
3623:
3611:
3588:
3576:
3560:
3542:
3416:
3402:
3321:
3307:
3160:
3157:
3151:
3148:
3115:
3109:
3070:
3067:
3061:
3058:
3024:
3018:
2985:
2979:
2963:
2960:
2954:
2951:
2897:
2891:
2849:
2843:
2799:{\displaystyle \mathbb {Z} /p}
2630:
2624:
2602:
2590:
2584:
2578:
2575:
2569:
2461:means a submanifold that is a
2418:
2406:
2393:
2390:
2378:
2326:
2292:
2280:
2258:
2246:
2208:
2198:
2151:
2139:
2113:
2101:
2045:
2033:
2014:
2011:
1999:
1983:
1971:
1867:
1855:
1810:
1798:
1761:
1749:
1709:
1706:
1697:
1683:
1670:
1652:
1649:
1637:
1624:
1621:
1612:
1598:
1576:
1553:
1514:
1511:
1499:
1480:
1477:
1471:
1458:
1455:
1449:
1436:
1433:
1421:
1408:
1374:
1356:
1317:
1314:
1308:
1289:
1286:
1274:
1261:
1258:
1252:
1236:
1230:
1217:
1214:
1208:
1195:
1123:
1117:
1104:
1101:
1095:
1048:
1042:
1029:
1026:
1020:
970:
723:
677:
636:
608:
562:
469:
450:
323:
285:
246:
223:
13:
1:
10183:
9649:(complex periodic K-theory),
9388:Various different flavors of
8966:is the union of subcomplexes
8681:generalized cohomology theory
8389:is the union of subcomplexes
3965:. It follows that for a ring
3765:is the class of a hyperplane
3467:= −, by graded-commutativity.
3395:) in the 2-dimensional torus
3356:denote a point in the circle
3232:, the cohomology ring of the
1838:, then the cohomology groups
1534:universal coefficient theorem
505:{\displaystyle \partial _{i}}
10526:
9921:
9613:(real connective K-theory),
7871:Alexander–Spanier cohomology
6524:, one has cohomology groups
6493:Alexander–Spanier cohomology
6440:by dualizing Čech homology.
5977:which is an isomorphism for
3201:, unless stated otherwise.
3124:{\displaystyle X-f^{-1}(N).}
2910:is to give a continuous map
2769:{\displaystyle \mathbb {Z} }
2473:is automatically compact if
2431:is a homomorphism of graded
1380:{\displaystyle H^{i}(X,Y;A)}
182:from the cohomology ring of
97:, and some kind of function
7:
10398:University of Chicago Press
10371:Encyclopedia of Mathematics
9909:
9469:{\displaystyle MSO^{*}(X),}
7930:generalized homology theory
7861:List of cohomology theories
6977:{\displaystyle \ell \neq p}
6081:A closed oriented manifold
5467:{\displaystyle \pi _{1}(X)}
5342:connected, it follows that
4937:{\displaystyle X\to K(A,j)}
3835:in degree 1, and the class
3711:/2 spanned by the elements
3455: := in degree 1, and
3428:{\displaystyle (S^{1})^{2}}
3333:{\displaystyle (S^{1})^{n}}
3297:The cohomology ring of the
3192:
2702:with an orientation on its
10:
10922:
10773:Banach fixed-point theorem
10341:Cambridge University Press
10273:Princeton University Press
9574:(real periodic K-theory),
9513:{\displaystyle MU^{*}(X),}
7858:
7435:{\displaystyle Z\subset X}
6638:of the left exact functor
6502:
6484:defined sheaf cohomology.
6012:), and a compact oriented
5614:For any topological space
5607:
5377:{\displaystyle H^{1}(X,A)}
5307:{\displaystyle H^{1}(X,A)}
5254:{\displaystyle X\to S^{1}}
4764:
4730:vanishes on a codimension-
4662:
4651:is a perfect pairing over
4444:
4050:). Namely, for any spaces
3799:≥ 0 has a basis as a free
3435:. Then the cohomology of (
1873:{\displaystyle H^{i}(X,A)}
1816:{\displaystyle H_{i}(X,F)}
1767:{\displaystyle H^{i}(X,F)}
805:negative. The elements of
10806:
10763:
10727:
10613:
10602:
10534:
9813:Coherent sheaf cohomology
9778:Other cohomology theories
9692:Brown–Peterson cohomology
9681:{\displaystyle ku^{*}(X)}
9606:{\displaystyle ko^{*}(X)}
9567:{\displaystyle KO^{*}(X)}
9424:{\displaystyle MO^{*}(X)}
9282:weak homotopy equivalence
8998:) induces an isomorphism
8421:) induces an isomorphism
7879:Eilenberg–Steenrod axioms
6984:. This is defined as the
6041:). The cap product ∩ ∈
6024:determines an element of
6016:-dimensional submanifold
5746:and any commutative ring
4680:over a topological space
3030:{\displaystyle f^{-1}(N)}
2998:as lying on the subspace
2922:and a closed codimension-
2081:. This product makes the
830:{\displaystyle C_{i}^{*}}
404:, and replace each group
400:Now fix an abelian group
126:gives rise to a function
10422:Switzer, Robert (1975),
9793:André–Quillen cohomology
9749:{\displaystyle E^{*}(X)}
9642:{\displaystyle K^{*}(X)}
9124:), then the inclusions (
8547:), then the inclusions (
7908:is a single point, then
7388:Weil cohomology theories
6712:-modules. Likewise, the
6421:At a 1935 conference in
4761:Eilenberg–MacLane spaces
4058:with cohomology classes
3744:complex projective space
2903:{\displaystyle H^{i}(X)}
2870:that can move freely on
2855:{\displaystyle H^{i}(X)}
979:{\displaystyle f:X\to Y}
205:For a topological space
9873:Intersection cohomology
6801:Cohomology of varieties
6767:) is isomorphic to Ext(
6751:on a topological space
6552:, the resulting groups
6520:on a topological space
5268:, say for a CW complex
5089:For example, the space
4903:by some continuous map
4824:Eilenberg–MacLane space
4771:For each abelian group
4767:Eilenberg–MacLane space
4751:Stiefel–Whitney classes
3742:The cohomology ring of
3641:The cohomology ring of
3228:For a positive integer
3166:{\displaystyle f^{*}()}
3076:{\displaystyle f^{*}()}
2646:where the intersection
2310:For any continuous map
1164:Mayer–Vietoris sequence
161:graded-commutative ring
54:, often defined from a
10828:Mathematics portal
10728:Metrics and properties
10714:Second-countable space
9898:Non-abelian cohomology
9883:Lie algebra cohomology
9833:Equivariant cohomology
9818:Crystalline cohomology
9750:
9682:
9643:
9607:
9568:
9514:
9470:
9425:
9380:
9341:stable homotopy groups
9333:
9242:
9082:
8951:
8738:,∅)). The axioms are:
8665:
8505:
8372:
8161:
8105:
8048:,∅)). The axioms are:
7974:natural transformation
7897:for smooth manifolds.
7893:for CW complexes, and
7843:
7719:
7696:
7529:
7436:
7410:
7409:{\displaystyle \geq 2}
7380:
7342:
7322:
7288:
7156:
6978:
6950:
6929:
6901:
6874:
6833:
6352:
6232:-dimensional manifold
6187:
5968:
5845:
5729:
5590:
5528:
5468:
5432:
5378:
5308:
5255:
5222:
5191:
5147:
5120:
5072:
5033:
4938:
4885:
4808:
4807:{\displaystyle K(A,j)}
4743:characteristic classes
4659:Characteristic classes
4590:
4432:
4236:
3683:is not orientable for
3633:
3429:
3377:
3334:
3256:
3226:
3167:
3125:
3077:
3031:
2992:
2904:
2856:
2800:
2770:
2726:For a smooth manifold
2640:
2489:manifold of dimension
2425:
2339:
2302:
2158:
2055:
1874:
1817:
1768:
1719:
1524:
1381:
1327:
1130:
1055:
980:
831:
733:
587:
506:
479:
333:
211:singular chain complex
9868:Hochschild cohomology
9751:
9683:
9644:
9608:
9569:
9515:
9471:
9426:
9381:
9334:
9243:
9083:
8974:, then the inclusion
8952:
8720:boundary homomorphism
8666:
8506:
8397:, then the inclusion
8373:
8162:
8106:
8034:) is a shorthand for
8018:boundary homomorphism
7883:simplicial cohomology
7844:
7720:
7697:
7530:
7437:
7411:
7381:
7343:
7341:{\displaystyle \ell }
7323:
7289:
7157:
6979:
6951:
6949:{\displaystyle \ell }
6930:
6909:Grothendieck topology
6902:
6875:
6834:
6586:holomorphic functions
6406:theorem; a result on
6365:on the cohomology of
6353:
6188:
5969:
5846:
5730:
5591:
5529:
5469:
5433:
5379:
5309:
5256:
5223:
5221:{\displaystyle S^{1}}
5192:
5148:
5146:{\displaystyle S^{1}}
5121:
5073:
5034:
4939:
4886:
4809:
4591:
4433:
4237:
3643:real projective space
3634:
3430:
3378:
3376:{\displaystyle S^{1}}
3335:
3257:
3255:{\displaystyle S^{n}}
3224:
3168:
3131:The cohomology class
3126:
3078:
3032:
2993:
2905:
2857:
2801:
2771:
2641:
2497:gives an isomorphism
2426:
2340:
2303:
2159:
2056:
1940:and commutative ring
1875:
1818:
1769:
1720:
1540:. Namely, there is a
1525:
1382:
1328:
1138:contravariant functor
1131:
1056:
981:
845:with coefficients in
832:
760:is defined to be ker(
756:with coefficients in
734:
588:
507:
480:
334:
144:, which gives them a
10783:Invariance of domain
10735:Euler characteristic
10709:Bundle (mathematics)
9724:
9712:A cohomology theory
9653:
9617:
9578:
9539:
9482:
9435:
9396:
9346:
9299:
9158:
9002:
8791:
8581:
8425:
8218:
8115:
8059:
7968:is a CW complex and
7887:simplicial complexes
7732:
7709:
7545:
7452:
7420:
7397:
7352:
7348:-adic cohomology of
7332:
7301:
7172:
6994:
6962:
6940:
6919:
6891:
6855:
6823:
6814:projective varieties
6376:-cochain on a space
6267:
6114:
5880:
5757:
5750:. The resulting map
5629:
5542:
5486:
5442:
5388:
5346:
5276:
5232:
5205:
5157:
5130:
5093:
5050:
4955:
4907:
4838:
4783:
4665:Characteristic class
4490:
4320:
4158:
4001:/2 coefficients) or
3529:
3470:More generally, let
3399:
3360:
3304:
3239:
3135:
3087:
3045:
3002:
2938:
2878:
2830:
2780:
2758:
2566:
2352:
2314:
2186:
2088:
1958:
1842:
1785:
1736:
1547:
1542:short exact sequence
1402:
1343:
1189:
1069:
994:
958:
849:. (Equivalently, an
809:
602:
519:
489:
418:
217:
66:in homology theory.
10906:Cohomology theories
10793:Tychonoff's theorem
10788:Poincaré conjecture
10542:General (point-set)
10087:, Proposition 3.38.
9703:elliptic cohomology
9701:Various flavors of
9524:, via a theorem of
9431:oriented cobordism
9363:
9316:
7922:George W. Whitehead
7891:cellular cohomology
7079:
6807:algebraic varieties
6669:to abelian groups.
6630:). This functor is
6596:homological algebra
6443:From 1936 to 1938,
6256: −
6226:intersection theory
6207:dual cell structure
5538:, with the element
4899:is the pullback of
4779:, there is a space
4775:and natural number
4479:) is isomorphic to
4098:) cohomology class
3652:/2 coefficients is
3439:) has a basis as a
3173:can move freely on
2812:Steenrod operations
2806:for a prime number
2441:homotopy equivalent
2182:in the sense that:
1572:
1337:relative cohomology
1183:long exact sequence
911:equivalence classes
869:.) Elements of ker(
826:
722:
672:
631:
579:
561:
435:
339:By definition, the
157:Singular cohomology
152:Singular cohomology
122:, composition with
10778:De Rham cohomology
10699:Polyhedral complex
10689:Simplicial complex
10475:10.1007/BF02566923
10336:Algebraic Topology
10302:Algebraic Geometry
9903:Quantum cohomology
9893:Motivic cohomology
9828:Deligne cohomology
9798:Bounded cohomology
9788:Algebraic K-theory
9746:
9678:
9639:
9603:
9564:
9510:
9466:
9421:
9376:
9349:
9329:
9302:
9238:
9198:
9078:
8947:
8661:
8593:
8501:
8368:
8157:
8101:
7920:≠ 0. Around 1960,
7895:de Rham cohomology
7839:
7715:
7705:If the subvariety
7692:
7525:
7523:
7432:
7406:
7376:
7338:
7318:
7284:
7152:
7062:
7058:
7043:
6974:
6946:
6935:one can construct
6925:
6897:
6870:
6829:
6574:algebraic geometry
6516:of abelian groups
6414:in terms of group
6408:topological groups
6404:Pontryagin duality
6348:
6183:
5964:
5841:
5725:
5622:is a bilinear map
5586:
5524:
5464:
5428:
5374:
5304:
5251:
5218:
5187:
5143:
5116:
5068:
5029:
4948:gives a bijection
4934:
4881:
4804:
4755:Pontryagin classes
4586:
4483:, and the product
4428:
4232:
3773:. More generally,
3761:in degree 2. Here
3668:is the class of a
3664:in degree 1. Here
3629:
3425:
3373:
3330:
3252:
3227:
3177:in the sense that
3163:
3121:
3073:
3027:
2988:
2900:
2852:
2796:
2766:
2748:differential forms
2636:
2452:closed submanifold
2421:
2335:
2298:
2180:graded-commutative
2154:
2128:
2051:
1926:finitely generated
1870:
1813:
1764:
1715:
1556:
1520:
1377:
1323:
1181:, then there is a
1126:
1061:on homology and a
1051:
976:
909:(because they are
907:cohomology classes
827:
812:
729:
702:
658:
611:
583:
565:
541:
502:
475:
421:
366:(called "singular
356:free abelian group
329:
69:From its start in
40:algebraic topology
34:, specifically in
10893:
10892:
10682:fundamental group
10298:Hartshorne, Robin
10261:Eilenberg, Samuel
10245:978-3-540-58660-9
9988:, pp. 62–63.
9878:Khovanov homology
9858:Galois cohomology
9823:Cyclic cohomology
9477:complex cobordism
9294:cohomotopy groups
9189:
9045:
8905:
8876:
8840:
8584:
8474:
8332:
8297:
8261:
7932:is a sequence of
7928:By definition, a
7718:{\displaystyle Z}
7278:
7036:
7034:
6928:{\displaystyle p}
6900:{\displaystyle p}
6832:{\displaystyle 0}
6582:regular functions
6427:Andrey Kolmogorov
6412:Alexander duality
6393:de Rham's theorem
6222:Solomon Lefschetz
6150:
6091:fundamental class
5984:For example, let
5738:for any integers
5476:fundamental group
5384:is isomorphic to
5001:
4828:classifying space
4669:An oriented real
4603:for each integer
4467:be a field. Then
2732:de Rham's theorem
2674:. In the case of
2119:
1904:is a commutative
1884:greater than the
1830:is a topological
1774:is precisely the
954:A continuous map
777:) and denoted by
699:
657:
652:
514:dual homomorphism
341:singular homology
306:
301:
268:
52:topological space
16:(Redirected from
10913:
10883:
10882:
10856:
10855:
10846:
10836:
10826:
10825:
10814:
10813:
10608:
10521:
10514:
10507:
10498:
10497:
10493:
10448:
10418:
10395:
10379:
10361:
10326:
10293:
10265:Steenrod, Norman
10256:
10223:
10200:
10178:
10172:
10166:
10165:
10154:
10148:
10142:
10136:
10130:
10124:
10123:, Section III.2.
10118:
10112:
10106:
10100:
10094:
10088:
10082:
10076:
10070:
10064:
10058:
10052:
10046:
10040:
10034:
10025:
10019:
10013:
10007:
10001:
10000:, Theorem II.29.
9995:
9989:
9983:
9977:
9971:
9965:
9959:
9953:
9943:
9937:
9931:
9888:Local cohomology
9863:Group cohomology
9838:Étale cohomology
9755:
9753:
9752:
9747:
9736:
9735:
9687:
9685:
9684:
9679:
9668:
9667:
9648:
9646:
9645:
9640:
9629:
9628:
9612:
9610:
9609:
9604:
9593:
9592:
9573:
9571:
9570:
9565:
9554:
9553:
9519:
9517:
9516:
9511:
9497:
9496:
9475:
9473:
9472:
9467:
9453:
9452:
9430:
9428:
9427:
9422:
9411:
9410:
9385:
9383:
9382:
9377:
9362:
9357:
9338:
9336:
9335:
9330:
9315:
9310:
9247:
9245:
9244:
9239:
9234:
9233:
9221:
9220:
9208:
9207:
9197:
9170:
9169:
9087:
9085:
9084:
9079:
9056:
9055:
9046:
9044:
9043:
9031:
9014:
9013:
8956:
8954:
8953:
8948:
8922:
8921:
8906:
8898:
8887:
8886:
8877:
8875:
8874:
8862:
8851:
8850:
8841:
8839:
8838:
8826:
8809:
8808:
8717:
8670:
8668:
8667:
8662:
8645:
8644:
8629:
8628:
8616:
8615:
8603:
8602:
8592:
8510:
8508:
8507:
8502:
8485:
8484:
8475:
8473:
8472:
8460:
8437:
8436:
8377:
8375:
8374:
8369:
8349:
8348:
8333:
8325:
8308:
8307:
8298:
8296:
8295:
8283:
8272:
8271:
8262:
8260:
8259:
8247:
8236:
8235:
8213:
8194:
8166:
8164:
8163:
8158:
8111:is homotopic to
8110:
8108:
8107:
8102:
8015:
7875:sheaf cohomology
7848:
7846:
7845:
7840:
7829:
7828:
7807:
7806:
7785:
7784:
7760:
7759:
7744:
7743:
7724:
7722:
7721:
7716:
7701:
7699:
7698:
7693:
7676:
7675:
7648:
7647:
7623:
7622:
7607:
7606:
7585:
7584:
7563:
7562:
7534:
7532:
7531:
7526:
7524:
7499:
7481:
7480:
7444:Cartesian square
7441:
7439:
7438:
7433:
7415:
7413:
7412:
7407:
7385:
7383:
7382:
7377:
7372:
7371:
7366:
7347:
7345:
7344:
7339:
7327:
7325:
7324:
7319:
7314:
7293:
7291:
7290:
7285:
7283:
7279:
7277:
7273:
7272:
7271:
7253:
7252:
7238:
7237:
7233:
7232:
7231:
7213:
7212:
7198:
7192:
7161:
7159:
7158:
7153:
7148:
7147:
7142:
7136:
7135:
7134:
7133:
7128:
7111:
7110:
7098:
7093:
7078:
7073:
7057:
7056:
7044:
7027:
7026:
7021:
7006:
7005:
6986:projective limit
6983:
6981:
6980:
6975:
6956:-adic cohomology
6955:
6953:
6952:
6947:
6934:
6932:
6931:
6926:
6906:
6904:
6903:
6898:
6879:
6877:
6876:
6871:
6869:
6868:
6863:
6845:Hodge structures
6838:
6836:
6835:
6830:
6816:over a field of
6663:derived category
6636:derived functors
6604:abelian category
6584:or the sheaf of
6578:complex analysis
6510:Sheaf cohomology
6505:Sheaf cohomology
6499:Sheaf cohomology
6464:Samuel Eilenberg
6357:
6355:
6354:
6349:
6335:
6334:
6301:
6300:
6279:
6278:
6246:general position
6192:
6190:
6189:
6184:
6167:
6166:
6151:
6143:
6126:
6125:
5973:
5971:
5970:
5965:
5936:
5935:
5920:
5919:
5892:
5891:
5850:
5848:
5847:
5842:
5825:
5824:
5797:
5796:
5769:
5768:
5734:
5732:
5731:
5726:
5709:
5708:
5675:
5674:
5647:
5646:
5595:
5593:
5592:
5587:
5579:
5574:
5560:
5559:
5533:
5531:
5530:
5525:
5517:
5512:
5498:
5497:
5473:
5471:
5470:
5465:
5454:
5453:
5437:
5435:
5434:
5429:
5409:
5408:
5383:
5381:
5380:
5375:
5358:
5357:
5313:
5311:
5310:
5305:
5288:
5287:
5260:
5258:
5257:
5252:
5250:
5249:
5227:
5225:
5224:
5219:
5217:
5216:
5196:
5194:
5193:
5188:
5183:
5169:
5168:
5152:
5150:
5149:
5144:
5142:
5141:
5125:
5123:
5122:
5117:
5106:
5077:
5075:
5074:
5071:{\displaystyle }
5069:
5042:for every space
5038:
5036:
5035:
5030:
5013:
5012:
5003:
5002:
5000:
4995:
4990:
4943:
4941:
4940:
4935:
4890:
4888:
4887:
4882:
4850:
4849:
4813:
4811:
4810:
4805:
4595:
4593:
4592:
4587:
4564:
4563:
4536:
4535:
4502:
4501:
4447:Poincaré duality
4441:Poincaré duality
4437:
4435:
4434:
4429:
4403:
4402:
4372:
4371:
4347:
4346:
4241:
4239:
4238:
4233:
4213:
4212:
4179:
4178:
4092:external product
3953:
3638:
3636:
3635:
3630:
3610:
3609:
3600:
3599:
3575:
3574:
3541:
3540:
3434:
3432:
3431:
3426:
3424:
3423:
3414:
3413:
3382:
3380:
3379:
3374:
3372:
3371:
3342:exterior algebra
3339:
3337:
3336:
3331:
3329:
3328:
3319:
3318:
3261:
3259:
3258:
3253:
3251:
3250:
3172:
3170:
3169:
3164:
3147:
3146:
3130:
3128:
3127:
3122:
3108:
3107:
3082:
3080:
3079:
3074:
3057:
3056:
3036:
3034:
3033:
3028:
3017:
3016:
2997:
2995:
2994:
2989:
2978:
2977:
2950:
2949:
2909:
2907:
2906:
2901:
2890:
2889:
2861:
2859:
2858:
2853:
2842:
2841:
2805:
2803:
2802:
2797:
2792:
2787:
2775:
2773:
2772:
2767:
2765:
2746:, defined using
2676:smooth manifolds
2645:
2643:
2642:
2637:
2623:
2622:
2495:Poincaré duality
2430:
2428:
2427:
2422:
2405:
2404:
2377:
2376:
2364:
2363:
2344:
2342:
2341:
2336:
2307:
2305:
2304:
2299:
2279:
2278:
2245:
2244:
2219:
2218:
2163:
2161:
2160:
2155:
2138:
2137:
2127:
2100:
2099:
2060:
2058:
2057:
2052:
2032:
2031:
1998:
1997:
1970:
1969:
1879:
1877:
1876:
1871:
1854:
1853:
1822:
1820:
1819:
1814:
1797:
1796:
1773:
1771:
1770:
1765:
1748:
1747:
1724:
1722:
1721:
1716:
1696:
1682:
1681:
1666:
1665:
1664:
1636:
1635:
1611:
1594:
1593:
1571:
1566:
1565:
1529:
1527:
1526:
1521:
1498:
1497:
1470:
1469:
1448:
1447:
1420:
1419:
1386:
1384:
1383:
1378:
1355:
1354:
1332:
1330:
1329:
1324:
1307:
1306:
1273:
1272:
1251:
1250:
1229:
1228:
1207:
1206:
1170:is the union of
1135:
1133:
1132:
1127:
1116:
1115:
1094:
1093:
1081:
1080:
1060:
1058:
1057:
1052:
1041:
1040:
1019:
1018:
1006:
1005:
985:
983:
982:
977:
926:commutative ring
836:
834:
833:
828:
825:
820:
750:cohomology group
738:
736:
735:
730:
721:
716:
701:
700:
698:
697:
696:
680:
675:
671:
666:
655:
654:
653:
651:
650:
649:
639:
634:
630:
625:
592:
590:
589:
584:
578:
573:
560:
555:
537:
536:
511:
509:
508:
503:
501:
500:
484:
482:
481:
476:
462:
461:
449:
434:
429:
338:
336:
335:
330:
322:
321:
304:
303:
302:
300:
299:
298:
288:
283:
280:
279:
270:
269:
267:
266:
265:
249:
244:
241:
240:
135:
121:
62:on the group of
21:
10921:
10920:
10916:
10915:
10914:
10912:
10911:
10910:
10896:
10895:
10894:
10889:
10820:
10802:
10798:Urysohn's lemma
10759:
10723:
10609:
10600:
10572:low-dimensional
10530:
10525:
10438:
10428:Springer-Verlag
10408:
10393:
10364:
10351:
10316:
10306:Springer-Verlag
10283:
10246:
10236:Springer-Verlag
10213:
10191:Dieudonné, Jean
10186:
10181:
10173:
10169:
10156:
10155:
10151:
10143:
10139:
10131:
10127:
10121:Hartshorne 1977
10119:
10115:
10111:, Section IV.3.
10107:
10103:
10095:
10091:
10083:
10079:
10071:
10067:
10059:
10055:
10047:
10043:
10039:, Theorem 3.19.
10035:
10028:
10024:, Theorem 3.15.
10020:
10016:
10012:, Example 3.16.
10008:
10004:
9996:
9992:
9984:
9980:
9976:, Theorem 3.11.
9972:
9968:
9960:
9956:
9948:, Theorem 3.5;
9944:
9940:
9932:
9928:
9924:
9912:
9907:
9848:Flat cohomology
9808:Čech cohomology
9803:BRST cohomology
9780:
9771:
9731:
9727:
9725:
9722:
9721:
9696:Morava K-theory
9663:
9659:
9654:
9651:
9650:
9624:
9620:
9618:
9615:
9614:
9588:
9584:
9579:
9576:
9575:
9549:
9545:
9540:
9537:
9536:
9492:
9488:
9483:
9480:
9479:
9448:
9444:
9436:
9433:
9432:
9406:
9402:
9397:
9394:
9393:
9358:
9353:
9347:
9344:
9343:
9311:
9306:
9300:
9297:
9296:
9229:
9225:
9216:
9212:
9203:
9199:
9193:
9165:
9161:
9159:
9156:
9155:
9141:
9132:
9123:
9114:
9051:
9047:
9039:
9035:
9030:
9009:
9005:
9003:
9000:
8999:
8911:
8907:
8897:
8882:
8878:
8870:
8866:
8861:
8846:
8842:
8834:
8830:
8825:
8804:
8800:
8792:
8789:
8788:
8692:
8640:
8636:
8624:
8620:
8611:
8607:
8598:
8594:
8588:
8582:
8579:
8578:
8564:
8555:
8546:
8537:
8480:
8476:
8468:
8464:
8459:
8432:
8428:
8426:
8423:
8422:
8338:
8334:
8324:
8303:
8299:
8291:
8287:
8282:
8267:
8263:
8255:
8251:
8246:
8231:
8227:
8219:
8216:
8215:
8196:
8182:
8116:
8113:
8112:
8060:
8057:
8056:
8043:
8029:
8009:
7991:
7982:
7976:
7943:
7902:dimension axiom
7867:Čech cohomology
7863:
7855:
7824:
7820:
7802:
7798:
7780:
7776:
7755:
7751:
7739:
7735:
7733:
7730:
7729:
7710:
7707:
7706:
7665:
7661:
7643:
7639:
7618:
7614:
7602:
7598:
7580:
7576:
7558:
7554:
7546:
7543:
7542:
7522:
7521:
7516:
7511:
7505:
7504:
7498:
7492:
7491:
7476:
7472:
7467:
7462:
7455:
7453:
7450:
7449:
7421:
7418:
7417:
7398:
7395:
7394:
7367:
7362:
7361:
7353:
7350:
7349:
7333:
7330:
7329:
7310:
7302:
7299:
7298:
7267:
7263:
7248:
7244:
7243:
7239:
7227:
7223:
7208:
7204:
7203:
7199:
7194:
7193:
7191:
7187:
7173:
7170:
7169:
7143:
7138:
7137:
7129:
7124:
7123:
7122:
7118:
7106:
7102:
7094:
7089:
7074:
7066:
7052:
7045:
7035:
7022:
7017:
7016:
7001:
6997:
6995:
6992:
6991:
6963:
6960:
6959:
6941:
6938:
6937:
6920:
6917:
6916:
6892:
6889:
6888:
6864:
6859:
6858:
6856:
6853:
6852:
6824:
6821:
6820:
6803:
6788:
6775:
6735:
6721:
6704:
6686:
6659:hypercohomology
6536:) for integers
6507:
6501:
6445:Hassler Whitney
6438:Čech cohomology
6434:Norman Steenrod
6389:Georges de Rham
6318:
6314:
6296:
6292:
6274:
6270:
6268:
6265:
6264:
6218:J. W. Alexander
6203:
6156:
6152:
6142:
6121:
6117:
6115:
6112:
6111:
6101:
6053:
6032:
5931:
5927:
5915:
5911:
5887:
5883:
5881:
5878:
5877:
5820:
5816:
5792:
5788:
5764:
5760:
5758:
5755:
5754:
5698:
5694:
5670:
5666:
5642:
5638:
5630:
5627:
5626:
5612:
5606:
5575:
5570:
5555:
5551:
5543:
5540:
5539:
5513:
5508:
5493:
5489:
5487:
5484:
5483:
5482:. For example,
5449:
5445:
5443:
5440:
5439:
5404:
5400:
5389:
5386:
5385:
5353:
5349:
5347:
5344:
5343:
5316:covering spaces
5283:
5279:
5277:
5274:
5273:
5245:
5241:
5233:
5230:
5229:
5212:
5208:
5206:
5203:
5202:
5179:
5164:
5160:
5158:
5155:
5154:
5137:
5133:
5131:
5128:
5127:
5102:
5094:
5091:
5090:
5051:
5048:
5047:
5008:
5004:
4996:
4991:
4989:
4988:
4956:
4953:
4952:
4908:
4905:
4904:
4895:on every space
4845:
4841:
4839:
4836:
4835:
4784:
4781:
4780:
4769:
4763:
4734:submanifold of
4667:
4661:
4635:with values in
4601:perfect pairing
4559:
4555:
4525:
4521:
4497:
4493:
4491:
4488:
4487:
4449:
4443:
4392:
4388:
4367:
4363:
4342:
4338:
4321:
4318:
4317:
4202:
4198:
4174:
4170:
4159:
4156:
4155:
4090:), there is an
4020:
3948:
3940:
3932:
3922:
3914:
3897:
3889:
3872:
3864:
3855:
3847:
3834:
3825:
3818:
3809:
3727:-cohomology of
3699:has an element
3695:-cohomology of
3605:
3601:
3595:
3591:
3570:
3566:
3536:
3532:
3530:
3527:
3526:
3504:Künneth formula
3419:
3415:
3409:
3405:
3400:
3397:
3396:
3367:
3363:
3361:
3358:
3357:
3324:
3320:
3314:
3310:
3305:
3302:
3301:
3276:polynomial ring
3246:
3242:
3240:
3237:
3236:
3195:
3142:
3138:
3136:
3133:
3132:
3100:
3096:
3088:
3085:
3084:
3052:
3048:
3046:
3043:
3042:
3009:
3005:
3003:
3000:
2999:
2973:
2969:
2945:
2941:
2939:
2936:
2935:
2885:
2881:
2879:
2876:
2875:
2837:
2833:
2831:
2828:
2827:
2788:
2783:
2781:
2778:
2777:
2761:
2759:
2756:
2755:
2612:
2608:
2567:
2564:
2563:
2558:that intersect
2516:
2448:closed manifold
2400:
2396:
2372:
2368:
2359:
2355:
2353:
2350:
2349:
2315:
2312:
2311:
2274:
2270:
2240:
2236:
2211:
2207:
2187:
2184:
2183:
2171:cohomology ring
2133:
2129:
2123:
2095:
2091:
2089:
2086:
2085:
2021:
2017:
1993:
1989:
1965:
1961:
1959:
1956:
1955:
1906:Noetherian ring
1849:
1845:
1843:
1840:
1839:
1792:
1788:
1786:
1783:
1782:
1743:
1739:
1737:
1734:
1733:
1692:
1677:
1673:
1660:
1659:
1655:
1631:
1627:
1607:
1583:
1579:
1567:
1561:
1560:
1548:
1545:
1544:
1487:
1483:
1465:
1461:
1443:
1439:
1415:
1411:
1403:
1400:
1399:
1350:
1346:
1344:
1341:
1340:
1296:
1292:
1268:
1264:
1246:
1242:
1224:
1220:
1202:
1198:
1190:
1187:
1186:
1111:
1107:
1089:
1085:
1076:
1072:
1070:
1067:
1066:
1036:
1032:
1014:
1010:
1001:
997:
995:
992:
991:
959:
956:
955:
821:
816:
810:
807:
806:
776:
765:
741:For an integer
717:
706:
686:
682:
681:
676:
674:
673:
667:
662:
645:
641:
640:
635:
633:
632:
626:
615:
603:
600:
599:
597:cochain complex
574:
569:
556:
545:
526:
522:
520:
517:
516:
496:
492:
490:
487:
486:
457:
453:
439:
430:
425:
419:
416:
415:
409:
392:
379:
352:
311:
307:
294:
290:
289:
284:
282:
281:
275:
271:
255:
251:
250:
245:
243:
242:
230:
226:
218:
215:
214:
200:homotopy groups
154:
127:
109:
56:cochain complex
36:homology theory
28:
23:
22:
15:
12:
11:
5:
10919:
10909:
10908:
10891:
10890:
10888:
10887:
10877:
10876:
10875:
10870:
10865:
10850:
10840:
10830:
10818:
10807:
10804:
10803:
10801:
10800:
10795:
10790:
10785:
10780:
10775:
10769:
10767:
10761:
10760:
10758:
10757:
10752:
10747:
10745:Winding number
10742:
10737:
10731:
10729:
10725:
10724:
10722:
10721:
10716:
10711:
10706:
10701:
10696:
10691:
10686:
10685:
10684:
10679:
10677:homotopy group
10669:
10668:
10667:
10662:
10657:
10652:
10647:
10637:
10632:
10627:
10617:
10615:
10611:
10610:
10603:
10601:
10599:
10598:
10593:
10588:
10587:
10586:
10576:
10575:
10574:
10564:
10559:
10554:
10549:
10544:
10538:
10536:
10532:
10531:
10524:
10523:
10516:
10509:
10501:
10495:
10494:
10449:
10436:
10419:
10406:
10381:
10362:
10349:
10331:Hatcher, Allen
10327:
10314:
10294:
10281:
10257:
10244:
10228:Dold, Albrecht
10224:
10211:
10185:
10182:
10180:
10179:
10167:
10149:
10137:
10125:
10113:
10109:Dieudonné 1989
10101:
10099:, p. 177.
10089:
10077:
10075:, p. 186.
10065:
10063:, Example 3.7.
10053:
10051:, p. 222.
10041:
10026:
10014:
10002:
9990:
9978:
9966:
9954:
9946:Hatcher (2001)
9938:
9936:, p. 108.
9925:
9923:
9920:
9919:
9918:
9911:
9908:
9906:
9905:
9900:
9895:
9890:
9885:
9880:
9875:
9870:
9865:
9860:
9855:
9853:Floer homology
9850:
9845:
9840:
9835:
9830:
9825:
9820:
9815:
9810:
9805:
9800:
9795:
9790:
9784:
9779:
9776:
9769:
9745:
9742:
9739:
9734:
9730:
9718:multiplicative
9716:is said to be
9707:
9706:
9699:
9689:
9677:
9674:
9671:
9666:
9662:
9658:
9638:
9635:
9632:
9627:
9623:
9602:
9599:
9596:
9591:
9587:
9583:
9563:
9560:
9557:
9552:
9548:
9544:
9529:
9526:Daniel Quillen
9509:
9506:
9503:
9500:
9495:
9491:
9487:
9465:
9462:
9459:
9456:
9451:
9447:
9443:
9440:
9420:
9417:
9414:
9409:
9405:
9401:
9386:
9375:
9372:
9369:
9366:
9361:
9356:
9352:
9328:
9325:
9322:
9319:
9314:
9309:
9305:
9254:
9253:
9237:
9232:
9228:
9224:
9219:
9215:
9211:
9206:
9202:
9196:
9192:
9188:
9185:
9182:
9179:
9176:
9173:
9168:
9164:
9137:
9128:
9119:
9110:
9093:
9077:
9074:
9071:
9068:
9065:
9062:
9059:
9054:
9050:
9042:
9038:
9034:
9029:
9026:
9023:
9020:
9017:
9012:
9008:
8957:
8946:
8943:
8940:
8937:
8934:
8931:
8928:
8925:
8920:
8917:
8914:
8910:
8904:
8901:
8896:
8893:
8890:
8885:
8881:
8873:
8869:
8865:
8860:
8857:
8854:
8849:
8845:
8837:
8833:
8829:
8824:
8821:
8818:
8815:
8812:
8807:
8803:
8799:
8796:
8746:
8687:(for integers
8677:
8676:
8660:
8657:
8654:
8651:
8648:
8643:
8639:
8635:
8632:
8627:
8623:
8619:
8614:
8610:
8606:
8601:
8597:
8591:
8587:
8560:
8551:
8542:
8533:
8516:
8500:
8497:
8494:
8491:
8488:
8483:
8479:
8471:
8467:
8463:
8458:
8455:
8452:
8449:
8446:
8443:
8440:
8435:
8431:
8378:
8367:
8364:
8361:
8358:
8355:
8352:
8347:
8344:
8341:
8337:
8331:
8328:
8323:
8320:
8317:
8314:
8311:
8306:
8302:
8294:
8290:
8286:
8281:
8278:
8275:
8270:
8266:
8258:
8254:
8250:
8245:
8242:
8239:
8234:
8230:
8226:
8223:
8168:
8156:
8153:
8150:
8147:
8144:
8141:
8138:
8135:
8132:
8129:
8126:
8123:
8120:
8100:
8097:
8094:
8091:
8088:
8085:
8082:
8079:
8076:
8073:
8070:
8067:
8064:
8038:
8024:
8004:
7987:
7978:
7944:(for integers
7939:
7916:) = 0 for all
7854:
7851:
7850:
7849:
7838:
7835:
7832:
7827:
7823:
7819:
7816:
7813:
7810:
7805:
7801:
7797:
7794:
7791:
7788:
7783:
7779:
7775:
7772:
7769:
7766:
7763:
7758:
7754:
7750:
7747:
7742:
7738:
7714:
7703:
7702:
7691:
7688:
7685:
7682:
7679:
7674:
7671:
7668:
7664:
7660:
7657:
7654:
7651:
7646:
7642:
7638:
7635:
7632:
7629:
7626:
7621:
7617:
7613:
7610:
7605:
7601:
7597:
7594:
7591:
7588:
7583:
7579:
7575:
7572:
7569:
7566:
7561:
7557:
7553:
7550:
7536:
7535:
7520:
7517:
7515:
7512:
7510:
7507:
7506:
7503:
7500:
7497:
7494:
7493:
7490:
7487:
7484:
7479:
7475:
7471:
7468:
7466:
7463:
7461:
7458:
7457:
7431:
7428:
7425:
7405:
7402:
7375:
7370:
7365:
7360:
7357:
7337:
7317:
7313:
7309:
7306:
7295:
7294:
7282:
7276:
7270:
7266:
7262:
7259:
7256:
7251:
7247:
7242:
7236:
7230:
7226:
7222:
7219:
7216:
7211:
7207:
7202:
7197:
7190:
7186:
7183:
7180:
7177:
7163:
7162:
7151:
7146:
7141:
7132:
7127:
7121:
7117:
7114:
7109:
7105:
7101:
7097:
7092:
7088:
7085:
7082:
7077:
7072:
7069:
7065:
7061:
7055:
7051:
7048:
7042:
7039:
7033:
7030:
7025:
7020:
7015:
7012:
7009:
7004:
7000:
6973:
6970:
6967:
6945:
6924:
6913:étale topology
6896:
6867:
6862:
6828:
6818:characteristic
6802:
6799:
6784:
6771:
6731:
6717:
6700:
6682:
6665:of sheaves on
6606:of sheaves on
6542:constant sheaf
6503:Main article:
6500:
6497:
6451:developed the
6400:Lev Pontryagin
6359:
6358:
6347:
6344:
6341:
6338:
6333:
6330:
6327:
6324:
6321:
6317:
6313:
6310:
6307:
6304:
6299:
6295:
6291:
6288:
6285:
6282:
6277:
6273:
6211:Henri Poincaré
6202:
6199:
6182:
6179:
6176:
6173:
6170:
6165:
6162:
6159:
6155:
6149:
6146:
6141:
6138:
6135:
6132:
6129:
6124:
6120:
6097:
6045:
6028:
5975:
5974:
5963:
5960:
5957:
5954:
5951:
5948:
5945:
5942:
5939:
5934:
5930:
5926:
5923:
5918:
5914:
5910:
5907:
5904:
5901:
5898:
5895:
5890:
5886:
5852:
5851:
5840:
5837:
5834:
5831:
5828:
5823:
5819:
5815:
5812:
5809:
5806:
5803:
5800:
5795:
5791:
5787:
5784:
5781:
5778:
5775:
5772:
5767:
5763:
5736:
5735:
5724:
5721:
5718:
5715:
5712:
5707:
5704:
5701:
5697:
5693:
5690:
5687:
5684:
5681:
5678:
5673:
5669:
5665:
5662:
5659:
5656:
5653:
5650:
5645:
5641:
5637:
5634:
5608:Main article:
5605:
5602:
5585:
5582:
5578:
5573:
5569:
5566:
5563:
5558:
5554:
5550:
5547:
5523:
5520:
5516:
5511:
5507:
5504:
5501:
5496:
5492:
5463:
5460:
5457:
5452:
5448:
5427:
5424:
5421:
5418:
5415:
5412:
5407:
5403:
5399:
5396:
5393:
5373:
5370:
5367:
5364:
5361:
5356:
5352:
5326:, also called
5303:
5300:
5297:
5294:
5291:
5286:
5282:
5248:
5244:
5240:
5237:
5215:
5211:
5201:of a point on
5186:
5182:
5178:
5175:
5172:
5167:
5163:
5140:
5136:
5115:
5112:
5109:
5105:
5101:
5098:
5067:
5064:
5061:
5058:
5055:
5040:
5039:
5028:
5025:
5022:
5019:
5016:
5011:
5007:
4999:
4994:
4987:
4984:
4981:
4978:
4975:
4972:
4969:
4966:
4963:
4960:
4933:
4930:
4927:
4924:
4921:
4918:
4915:
4912:
4880:
4877:
4874:
4871:
4868:
4865:
4862:
4859:
4856:
4853:
4848:
4844:
4803:
4800:
4797:
4794:
4791:
4788:
4765:Main article:
4762:
4759:
4663:Main article:
4660:
4657:
4597:
4596:
4585:
4582:
4579:
4576:
4573:
4570:
4567:
4562:
4558:
4554:
4551:
4548:
4545:
4542:
4539:
4534:
4531:
4528:
4524:
4520:
4517:
4514:
4511:
4508:
4505:
4500:
4496:
4445:Main article:
4442:
4439:
4427:
4424:
4421:
4418:
4415:
4412:
4409:
4406:
4401:
4398:
4395:
4391:
4387:
4384:
4381:
4378:
4375:
4370:
4366:
4362:
4359:
4356:
4353:
4350:
4345:
4341:
4337:
4334:
4331:
4328:
4325:
4231:
4228:
4225:
4222:
4219:
4216:
4211:
4208:
4205:
4201:
4197:
4194:
4191:
4188:
4185:
4182:
4177:
4173:
4169:
4166:
4163:
4019:
4016:
4015:
4014:
4013:coefficients).
3955:
3944:
3936:
3918:
3910:
3893:
3885:
3868:
3860:
3851:
3843:
3830:
3823:
3814:
3807:
3786:
3740:
3639:
3628:
3625:
3622:
3619:
3616:
3613:
3608:
3604:
3598:
3594:
3590:
3587:
3584:
3581:
3578:
3573:
3569:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3544:
3539:
3535:
3519:tensor product
3468:
3451: := and
3422:
3418:
3412:
3408:
3404:
3370:
3366:
3327:
3323:
3317:
3313:
3309:
3295:
3249:
3245:
3218:
3194:
3191:
3162:
3159:
3156:
3153:
3150:
3145:
3141:
3120:
3117:
3114:
3111:
3106:
3103:
3099:
3095:
3092:
3072:
3069:
3066:
3063:
3060:
3055:
3051:
3026:
3023:
3020:
3015:
3012:
3008:
2987:
2984:
2981:
2976:
2972:
2968:
2965:
2962:
2959:
2956:
2953:
2948:
2944:
2918:to a manifold
2899:
2896:
2893:
2888:
2884:
2851:
2848:
2845:
2840:
2836:
2826:, elements of
2820:
2819:
2795:
2791:
2786:
2764:
2752:chain homotopy
2724:
2635:
2632:
2629:
2626:
2621:
2618:
2615:
2611:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2508:
2457:of a manifold
2420:
2417:
2414:
2411:
2408:
2403:
2399:
2395:
2392:
2389:
2386:
2383:
2380:
2375:
2371:
2367:
2362:
2358:
2334:
2331:
2328:
2325:
2322:
2319:
2297:
2294:
2291:
2288:
2285:
2282:
2277:
2273:
2269:
2266:
2263:
2260:
2257:
2254:
2251:
2248:
2243:
2239:
2235:
2232:
2228:
2225:
2222:
2217:
2214:
2210:
2206:
2203:
2200:
2197:
2194:
2191:
2153:
2150:
2147:
2144:
2141:
2136:
2132:
2126:
2122:
2118:
2115:
2112:
2109:
2106:
2103:
2098:
2094:
2069:is written as
2050:
2047:
2044:
2041:
2038:
2035:
2030:
2027:
2024:
2020:
2016:
2013:
2010:
2007:
2004:
2001:
1996:
1992:
1988:
1985:
1982:
1979:
1976:
1973:
1968:
1964:
1934:
1933:
1869:
1866:
1863:
1860:
1857:
1852:
1848:
1824:
1812:
1809:
1806:
1803:
1800:
1795:
1791:
1763:
1760:
1757:
1754:
1751:
1746:
1742:
1714:
1711:
1708:
1705:
1702:
1699:
1695:
1691:
1688:
1685:
1680:
1676:
1672:
1669:
1663:
1658:
1654:
1651:
1648:
1645:
1642:
1639:
1634:
1630:
1626:
1623:
1620:
1617:
1614:
1610:
1606:
1603:
1600:
1597:
1592:
1589:
1586:
1582:
1578:
1575:
1570:
1564:
1559:
1555:
1552:
1530:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1496:
1493:
1490:
1486:
1482:
1479:
1476:
1473:
1468:
1464:
1460:
1457:
1454:
1451:
1446:
1442:
1438:
1435:
1432:
1429:
1426:
1423:
1418:
1414:
1410:
1407:
1376:
1373:
1370:
1367:
1364:
1361:
1358:
1353:
1349:
1333:
1322:
1319:
1316:
1313:
1310:
1305:
1302:
1299:
1295:
1291:
1288:
1285:
1282:
1279:
1276:
1271:
1267:
1263:
1260:
1257:
1254:
1249:
1245:
1241:
1238:
1235:
1232:
1227:
1223:
1219:
1216:
1213:
1210:
1205:
1201:
1197:
1194:
1160:
1145:
1125:
1122:
1119:
1114:
1110:
1106:
1103:
1100:
1097:
1092:
1088:
1084:
1079:
1075:
1050:
1047:
1044:
1039:
1035:
1031:
1028:
1025:
1022:
1017:
1013:
1009:
1004:
1000:
975:
972:
969:
966:
963:
913:of cocycles).
861:-simplices in
824:
819:
815:
801:) is zero for
771:
763:
728:
725:
720:
715:
712:
709:
705:
695:
692:
689:
685:
679:
670:
665:
661:
648:
644:
638:
629:
624:
621:
618:
614:
610:
607:
582:
577:
572:
568:
564:
559:
554:
551:
548:
544:
540:
535:
532:
529:
525:
499:
495:
474:
471:
468:
465:
460:
456:
452:
448:
445:
442:
438:
433:
428:
424:
407:
388:
375:
370:-simplices in
350:
328:
325:
320:
317:
314:
310:
297:
293:
287:
278:
274:
264:
261:
258:
254:
248:
239:
236:
233:
229:
225:
222:
165:continuous map
153:
150:
48:abelian groups
26:
9:
6:
4:
3:
2:
10918:
10907:
10904:
10903:
10901:
10886:
10878:
10874:
10871:
10869:
10866:
10864:
10861:
10860:
10859:
10851:
10849:
10845:
10841:
10839:
10835:
10831:
10829:
10824:
10819:
10817:
10809:
10808:
10805:
10799:
10796:
10794:
10791:
10789:
10786:
10784:
10781:
10779:
10776:
10774:
10771:
10770:
10768:
10766:
10762:
10756:
10755:Orientability
10753:
10751:
10748:
10746:
10743:
10741:
10738:
10736:
10733:
10732:
10730:
10726:
10720:
10717:
10715:
10712:
10710:
10707:
10705:
10702:
10700:
10697:
10695:
10692:
10690:
10687:
10683:
10680:
10678:
10675:
10674:
10673:
10670:
10666:
10663:
10661:
10658:
10656:
10653:
10651:
10648:
10646:
10643:
10642:
10641:
10638:
10636:
10633:
10631:
10628:
10626:
10622:
10619:
10618:
10616:
10612:
10607:
10597:
10594:
10592:
10591:Set-theoretic
10589:
10585:
10582:
10581:
10580:
10577:
10573:
10570:
10569:
10568:
10565:
10563:
10560:
10558:
10555:
10553:
10552:Combinatorial
10550:
10548:
10545:
10543:
10540:
10539:
10537:
10533:
10529:
10522:
10517:
10515:
10510:
10508:
10503:
10502:
10499:
10492:
10488:
10484:
10480:
10476:
10472:
10468:
10464:
10463:
10458:
10454:
10450:
10447:
10443:
10439:
10437:3-540-42750-3
10433:
10429:
10425:
10420:
10417:
10413:
10409:
10407:0-226-51182-0
10403:
10399:
10392:
10391:
10386:
10385:May, J. Peter
10382:
10377:
10373:
10372:
10367:
10363:
10360:
10356:
10352:
10350:0-521-79540-0
10346:
10342:
10338:
10337:
10332:
10328:
10325:
10321:
10317:
10315:0-387-90244-9
10311:
10307:
10303:
10299:
10295:
10292:
10288:
10284:
10282:9780691627236
10278:
10274:
10270:
10266:
10262:
10258:
10255:
10251:
10247:
10241:
10237:
10233:
10229:
10225:
10222:
10218:
10214:
10212:0-8176-3388-X
10208:
10204:
10199:
10198:
10192:
10188:
10187:
10176:
10171:
10163:
10159:
10153:
10146:
10141:
10135:, p. 95.
10134:
10129:
10122:
10117:
10110:
10105:
10098:
10093:
10086:
10081:
10074:
10069:
10062:
10057:
10050:
10045:
10038:
10033:
10031:
10023:
10018:
10011:
10006:
9999:
9994:
9987:
9982:
9975:
9970:
9963:
9958:
9951:
9947:
9942:
9935:
9930:
9926:
9917:
9914:
9913:
9904:
9901:
9899:
9896:
9894:
9891:
9889:
9886:
9884:
9881:
9879:
9876:
9874:
9871:
9869:
9866:
9864:
9861:
9859:
9856:
9854:
9851:
9849:
9846:
9844:
9841:
9839:
9836:
9834:
9831:
9829:
9826:
9824:
9821:
9819:
9816:
9814:
9811:
9809:
9806:
9804:
9801:
9799:
9796:
9794:
9791:
9789:
9786:
9785:
9783:
9775:
9773:
9772:ring spectrum
9768:
9764:, such as an
9763:
9762:ring spectrum
9759:
9740:
9732:
9728:
9719:
9715:
9710:
9704:
9700:
9697:
9693:
9690:
9672:
9664:
9660:
9656:
9633:
9625:
9621:
9597:
9589:
9585:
9581:
9558:
9550:
9546:
9542:
9534:
9530:
9527:
9523:
9522:formal groups
9507:
9501:
9493:
9489:
9485:
9478:
9463:
9457:
9449:
9445:
9441:
9438:
9415:
9407:
9403:
9399:
9391:
9387:
9373:
9367:
9359:
9354:
9350:
9342:
9326:
9320:
9312:
9307:
9303:
9295:
9291:
9290:
9289:
9286:
9283:
9278:
9276:
9272:
9266:
9263:
9259:
9251:
9230:
9226:
9222:
9217:
9213:
9204:
9200:
9194:
9190:
9180:
9177:
9174:
9166:
9162:
9153:
9152:product group
9149:
9145:
9140:
9136:
9131:
9127:
9122:
9118:
9113:
9109:
9105:
9101:
9097:
9094:
9091:
9072:
9069:
9066:
9063:
9060:
9052:
9048:
9040:
9036:
9024:
9021:
9018:
9010:
9006:
8997:
8993:
8989:
8985:
8981:
8977:
8973:
8969:
8965:
8961:
8958:
8944:
8941:
8932:
8929:
8926:
8918:
8915:
8912:
8908:
8902:
8891:
8883:
8879:
8871:
8867:
8855:
8847:
8843:
8835:
8831:
8819:
8816:
8813:
8805:
8801:
8794:
8786:
8782:
8778:
8774:
8770:
8766:
8762:
8758:
8754:
8751:: Each pair (
8750:
8747:
8744:
8741:
8740:
8739:
8737:
8733:
8729:
8725:
8721:
8715:
8711:
8707:
8703:
8699:
8695:
8690:
8686:
8682:
8674:
8655:
8652:
8649:
8641:
8637:
8625:
8621:
8617:
8612:
8608:
8599:
8595:
8589:
8585:
8576:
8572:
8568:
8563:
8559:
8554:
8550:
8545:
8541:
8536:
8532:
8528:
8524:
8520:
8517:
8514:
8495:
8492:
8489:
8481:
8477:
8469:
8465:
8453:
8450:
8447:
8444:
8441:
8433:
8429:
8420:
8416:
8412:
8408:
8404:
8400:
8396:
8392:
8388:
8384:
8383:
8379:
8365:
8362:
8353:
8345:
8342:
8339:
8335:
8318:
8315:
8312:
8304:
8300:
8292:
8288:
8276:
8268:
8264:
8256:
8252:
8240:
8232:
8228:
8221:
8211:
8207:
8203:
8199:
8193:
8189:
8185:
8180:
8176:
8173:: Each pair (
8172:
8169:
8151:
8148:
8145:
8133:
8130:
8127:
8121:
8118:
8095:
8092:
8089:
8077:
8074:
8071:
8065:
8062:
8054:
8051:
8050:
8049:
8047:
8041:
8037:
8033:
8027:
8023:
8019:
8013:
8007:
8003:
7999:
7995:
7990:
7986:
7981:
7975:
7971:
7967:
7963:
7959:
7955:
7951:
7947:
7942:
7938:
7935:
7931:
7926:
7923:
7919:
7915:
7911:
7907:
7903:
7898:
7896:
7892:
7888:
7884:
7880:
7876:
7872:
7868:
7862:
7857:
7833:
7825:
7821:
7817:
7811:
7803:
7799:
7795:
7789:
7781:
7777:
7773:
7764:
7756:
7752:
7748:
7740:
7736:
7728:
7727:
7726:
7712:
7689:
7680:
7672:
7669:
7666:
7662:
7652:
7644:
7640:
7627:
7619:
7615:
7611:
7603:
7599:
7595:
7589:
7581:
7577:
7567:
7559:
7555:
7548:
7541:
7540:
7539:
7518:
7508:
7485:
7477:
7473:
7469:
7459:
7448:
7447:
7446:
7445:
7429:
7426:
7423:
7403:
7400:
7391:
7389:
7368:
7355:
7335:
7304:
7280:
7274:
7268:
7264:
7260:
7257:
7254:
7249:
7245:
7240:
7234:
7228:
7224:
7220:
7217:
7214:
7209:
7205:
7200:
7188:
7184:
7181:
7178:
7175:
7168:
7167:
7166:
7149:
7144:
7130:
7119:
7107:
7103:
7095:
7086:
7083:
7075:
7070:
7067:
7063:
7059:
7049:
7046:
7040:
7037:
7031:
7023:
7013:
7010:
7002:
6998:
6990:
6989:
6988:
6987:
6971:
6968:
6965:
6957:
6943:
6922:
6914:
6910:
6894:
6886:
6881:
6865:
6850:
6846:
6842:
6839:. Tools from
6826:
6819:
6815:
6812:
6808:
6798:
6796:
6792:
6787:
6783:
6779:
6774:
6770:
6766:
6762:
6758:
6754:
6750:
6745:
6743:
6739:
6734:
6729:
6725:
6720:
6715:
6711:
6707:
6703:
6698:
6694:
6690:
6685:
6680:
6676:
6670:
6668:
6664:
6660:
6656:
6651:
6649:
6645:
6641:
6637:
6633:
6629:
6625:
6621:
6617:
6613:
6609:
6605:
6601:
6597:
6593:
6589:
6587:
6583:
6579:
6575:
6571:
6567:
6563:
6559:
6555:
6551:
6547:
6543:
6539:
6535:
6531:
6527:
6523:
6519:
6515:
6511:
6506:
6496:
6494:
6490:
6489:Edwin Spanier
6485:
6483:
6478:
6476:
6472:
6467:
6465:
6460:
6458:
6454:
6450:
6446:
6441:
6439:
6435:
6430:
6428:
6424:
6419:
6417:
6413:
6409:
6405:
6401:
6396:
6394:
6390:
6385:
6383:
6379:
6375:
6370:
6368:
6364:
6345:
6339:
6331:
6328:
6325:
6322:
6319:
6315:
6305:
6297:
6293:
6289:
6283:
6275:
6271:
6263:
6262:
6261:
6259:
6255:
6252: +
6251:
6247:
6243:
6240:-cycle and a
6239:
6235:
6231:
6227:
6223:
6219:
6214:
6212:
6208:
6198:
6196:
6177:
6174:
6171:
6163:
6160:
6157:
6153:
6147:
6136:
6133:
6130:
6122:
6118:
6109:
6105:
6100:
6096:
6092:
6088:
6085:of dimension
6084:
6079:
6077:
6073:
6069:
6065:
6061:
6057:
6052:
6048:
6044:
6040:
6036:
6031:
6027:
6023:
6019:
6015:
6011:
6007:
6003:
5999:
5995:
5991:
5987:
5982:
5980:
5961:
5955:
5952:
5946:
5943:
5940:
5932:
5928:
5921:
5916:
5912:
5902:
5899:
5896:
5888:
5884:
5876:
5875:
5874:
5872:
5868:
5863:
5861:
5857:
5835:
5832:
5829:
5821:
5817:
5807:
5804:
5801:
5793:
5789:
5785:
5779:
5776:
5773:
5765:
5761:
5753:
5752:
5751:
5749:
5745:
5741:
5719:
5716:
5713:
5705:
5702:
5699:
5695:
5685:
5682:
5679:
5671:
5667:
5663:
5657:
5654:
5651:
5643:
5639:
5635:
5632:
5625:
5624:
5623:
5621:
5617:
5611:
5601:
5599:
5580:
5576:
5567:
5564:
5556:
5552:
5548:
5545:
5537:
5518:
5514:
5505:
5502:
5494:
5490:
5481:
5477:
5458:
5450:
5446:
5422:
5419:
5413:
5405:
5401:
5394:
5391:
5368:
5365:
5362:
5354:
5350:
5341:
5337:
5333:
5331:
5325:
5321:
5317:
5298:
5295:
5292:
5284:
5280:
5271:
5267:
5262:
5246:
5242:
5235:
5213:
5209:
5200:
5176:
5173:
5165:
5161:
5138:
5134:
5110:
5107:
5096:
5087:
5085:
5081:
5062:
5059:
5056:
5045:
5023:
5020:
5017:
5009:
5005:
4997:
4979:
4976:
4973:
4967:
4964:
4961:
4951:
4950:
4949:
4947:
4928:
4925:
4922:
4916:
4910:
4902:
4898:
4894:
4875:
4872:
4866:
4863:
4860:
4854:
4846:
4842:
4833:
4829:
4825:
4821:
4817:
4798:
4795:
4792:
4786:
4778:
4774:
4768:
4758:
4756:
4752:
4748:
4747:Chern classes
4744:
4739:
4737:
4733:
4729:
4725:
4721:
4717:
4713:
4709:
4705:
4701:
4697:
4693:
4692:
4687:
4683:
4679:
4675:
4672:
4671:vector bundle
4666:
4656:
4654:
4650:
4646:
4642:
4638:
4634:
4630:
4626:
4622:
4618:
4614:
4610:
4606:
4602:
4583:
4580:
4574:
4571:
4568:
4560:
4556:
4546:
4543:
4540:
4532:
4529:
4526:
4522:
4518:
4512:
4509:
4506:
4498:
4494:
4486:
4485:
4484:
4482:
4478:
4474:
4470:
4466:
4462:
4458:
4454:
4448:
4438:
4425:
4419:
4416:
4413:
4410:
4407:
4399:
4396:
4393:
4389:
4385:
4376:
4368:
4364:
4351:
4343:
4339:
4332:
4329:
4326:
4323:
4315:
4311:
4307:
4303:
4299:
4295:
4291:
4287:
4283:
4279:
4275:
4271:
4267:
4263:
4259:
4255:
4251:
4247:
4242:
4229:
4223:
4220:
4217:
4209:
4206:
4203:
4199:
4195:
4189:
4186:
4183:
4175:
4167:
4164:
4161:
4153:
4149:
4145:
4141:
4137:
4133:
4129:
4125:
4121:
4117:
4113:
4109:
4105:
4101:
4097:
4096:cross product
4093:
4089:
4085:
4081:
4077:
4073:
4069:
4065:
4061:
4057:
4053:
4049:
4045:
4041:
4037:
4033:
4029:
4025:
4012:
4008:
4004:
4000:
3996:
3992:
3988:
3984:
3980:
3976:
3972:
3968:
3964:
3960:
3956:
3952:
3947:
3943:
3939:
3935:
3930:
3926:
3921:
3917:
3913:
3909:
3905:
3901:
3896:
3892:
3888:
3884:
3880:
3876:
3871:
3867:
3863:
3859:
3854:
3850:
3846:
3842:
3838:
3833:
3829:
3822:
3817:
3813:
3806:
3802:
3798:
3795:
3791:
3787:
3784:
3780:
3776:
3772:
3768:
3764:
3760:
3756:
3752:
3748:
3745:
3741:
3738:
3734:
3730:
3726:
3722:
3718:
3714:
3710:
3706:
3702:
3698:
3694:
3690:
3686:
3682:
3678:
3674:
3671:
3667:
3663:
3659:
3655:
3651:
3647:
3644:
3640:
3626:
3620:
3617:
3614:
3606:
3602:
3596:
3592:
3585:
3582:
3579:
3571:
3567:
3563:
3557:
3554:
3551:
3548:
3545:
3537:
3533:
3524:
3520:
3516:
3512:
3509:
3508:product space
3505:
3501:
3497:
3493:
3489:
3485:
3481:
3477:
3473:
3469:
3466:
3462:
3458:
3454:
3450:
3446:
3444:
3438:
3420:
3410:
3406:
3394:
3390:
3386:
3368:
3364:
3355:
3351:
3347:
3343:
3325:
3315:
3311:
3300:
3296:
3293:
3289:
3285:
3281:
3278:by the given
3277:
3273:
3272:quotient ring
3269:
3265:
3247:
3243:
3235:
3231:
3223:
3219:
3216:
3212:
3208:
3204:
3203:
3202:
3200:
3190:
3188:
3184:
3180:
3176:
3154:
3143:
3139:
3118:
3112:
3104:
3101:
3097:
3093:
3090:
3064:
3053:
3049:
3040:
3021:
3013:
3010:
3006:
2982:
2974:
2970:
2966:
2957:
2946:
2942:
2933:
2929:
2925:
2921:
2917:
2913:
2894:
2886:
2882:
2873:
2869:
2866:subspaces of
2865:
2846:
2838:
2834:
2825:
2817:
2813:
2809:
2793:
2789:
2753:
2749:
2745:
2741:
2737:
2733:
2729:
2725:
2721:
2717:
2713:
2709:
2705:
2704:normal bundle
2701:
2697:
2693:
2689:
2685:
2681:
2677:
2673:
2669:
2665:
2661:
2657:
2653:
2649:
2633:
2627:
2619:
2616:
2613:
2609:
2605:
2599:
2596:
2593:
2587:
2581:
2572:
2561:
2557:
2553:
2549:
2545:
2541:
2538:
2534:
2530:
2527:
2523:
2519:
2515:
2511:
2507:
2503:
2500:
2496:
2492:
2488:
2484:
2480:
2479:
2478:
2476:
2472:
2468:
2464:
2463:closed subset
2460:
2456:
2453:
2449:
2444:
2442:
2438:
2434:
2415:
2412:
2409:
2401:
2397:
2387:
2384:
2381:
2373:
2369:
2365:
2360:
2356:
2348:
2332:
2329:
2323:
2320:
2317:
2308:
2295:
2289:
2286:
2283:
2275:
2271:
2267:
2264:
2261:
2255:
2252:
2249:
2241:
2237:
2233:
2230:
2226:
2223:
2220:
2215:
2212:
2204:
2201:
2195:
2192:
2189:
2181:
2177:
2173:
2172:
2168:, called the
2167:
2148:
2145:
2142:
2134:
2130:
2124:
2120:
2116:
2110:
2107:
2104:
2096:
2092:
2084:
2080:
2077:or simply as
2076:
2072:
2068:
2064:
2048:
2042:
2039:
2036:
2028:
2025:
2022:
2018:
2008:
2005:
2002:
1994:
1990:
1986:
1980:
1977:
1974:
1966:
1962:
1953:
1952:
1948:, called the
1947:
1944:, there is a
1943:
1939:
1931:
1927:
1923:
1919:
1915:
1911:
1907:
1903:
1899:
1895:
1891:
1887:
1883:
1880:are zero for
1864:
1861:
1858:
1850:
1846:
1837:
1833:
1829:
1825:
1807:
1804:
1801:
1793:
1789:
1781:
1777:
1758:
1755:
1752:
1744:
1740:
1731:
1728:
1712:
1703:
1700:
1689:
1686:
1678:
1674:
1667:
1656:
1646:
1643:
1640:
1632:
1628:
1618:
1615:
1604:
1601:
1595:
1590:
1587:
1584:
1573:
1568:
1557:
1550:
1543:
1539:
1535:
1531:
1517:
1508:
1505:
1502:
1494:
1491:
1488:
1484:
1474:
1466:
1462:
1452:
1444:
1440:
1430:
1427:
1424:
1416:
1412:
1405:
1397:
1393:
1390:
1371:
1368:
1365:
1362:
1359:
1351:
1347:
1338:
1334:
1320:
1311:
1303:
1300:
1297:
1293:
1283:
1280:
1277:
1269:
1265:
1255:
1247:
1243:
1239:
1233:
1225:
1221:
1211:
1203:
1199:
1192:
1184:
1180:
1176:
1173:
1169:
1165:
1161:
1158:
1154:
1150:
1146:
1143:
1139:
1120:
1112:
1108:
1098:
1090:
1086:
1082:
1077:
1073:
1065:homomorphism
1064:
1045:
1037:
1033:
1023:
1015:
1011:
1007:
1002:
998:
990:homomorphism
989:
986:determines a
973:
967:
964:
961:
953:
952:
951:
948:
946:
942:
938:
934:
930:
927:
923:
919:
914:
912:
908:
905:) are called
904:
900:
896:
892:
888:
884:
880:
877:) are called
876:
872:
868:
864:
860:
856:
852:
848:
844:
842:
822:
817:
813:
804:
800:
796:
792:
789:). The group
788:
784:
780:
774:
770:
766:
759:
755:
751:
748:
744:
739:
726:
718:
713:
710:
707:
703:
693:
690:
687:
683:
668:
663:
659:
646:
642:
627:
622:
619:
616:
612:
605:
598:
593:
580:
575:
570:
566:
557:
552:
549:
546:
542:
538:
533:
530:
527:
523:
515:
497:
472:
466:
463:
458:
454:
436:
431:
426:
422:
414:
410:
403:
398:
396:
393:are zero for
391:
387:
383:
378:
373:
369:
365:
361:
357:
353:
346:
342:
326:
318:
315:
312:
308:
295:
276:
272:
262:
259:
256:
237:
234:
231:
227:
220:
212:
208:
203:
201:
197:
193:
189:
185:
181:
178:determines a
177:
173:
169:
166:
162:
158:
149:
147:
143:
139:
134:
130:
125:
120:
116:
112:
108:
104:
100:
96:
92:
88:
84:
83:contravariant
80:
76:
72:
67:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
10885:Publications
10750:Chern number
10740:Betti number
10623: /
10614:Key concepts
10583:
10562:Differential
10466:
10460:
10423:
10389:
10369:
10366:"Cohomology"
10335:
10301:
10268:
10231:
10196:
10175:Switzer 1975
10170:
10162:MathOverflow
10161:
10152:
10145:Switzer 1975
10140:
10128:
10116:
10104:
10092:
10085:Hatcher 2001
10080:
10073:Hatcher 2001
10068:
10061:Hatcher 2001
10056:
10049:Hatcher 2001
10044:
10037:Hatcher 2001
10022:Hatcher 2001
10017:
10010:Hatcher 2001
10005:
9993:
9981:
9974:Hatcher 2001
9969:
9957:
9941:
9934:Hatcher 2001
9929:
9781:
9766:
9757:
9717:
9713:
9711:
9708:
9287:
9279:
9275:triangulated
9271:phantom maps
9267:
9255:
9249:
9147:
9143:
9138:
9134:
9129:
9125:
9120:
9116:
9111:
9107:
9103:
9099:
9095:
9089:
8995:
8991:
8987:
8983:
8979:
8975:
8971:
8967:
8963:
8959:
8784:
8780:
8776:
8772:
8768:
8764:
8760:
8756:
8752:
8748:
8742:
8735:
8731:
8727:
8723:
8719:
8713:
8709:
8705:
8701:
8697:
8693:
8688:
8684:
8680:
8678:
8672:
8570:
8566:
8561:
8557:
8552:
8548:
8543:
8539:
8534:
8530:
8526:
8522:
8518:
8512:
8418:
8414:
8410:
8406:
8402:
8398:
8394:
8390:
8386:
8380:
8209:
8205:
8201:
8197:
8191:
8187:
8183:
8178:
8174:
8170:
8052:
8045:
8039:
8035:
8031:
8025:
8021:
8017:
8011:
8005:
8001:
7997:
7993:
7988:
7984:
7979:
7969:
7965:
7961:
7957:
7945:
7940:
7936:
7929:
7927:
7917:
7913:
7909:
7905:
7901:
7899:
7864:
7856:
7704:
7537:
7392:
7296:
7164:
6885:finite field
6882:
6849:hypersurface
6841:Hodge theory
6804:
6794:
6790:
6785:
6781:
6777:
6772:
6768:
6764:
6760:
6756:
6752:
6748:
6746:
6741:
6737:
6732:
6727:
6723:
6718:
6709:
6705:
6701:
6696:
6692:
6688:
6683:
6674:
6671:
6666:
6654:
6652:
6647:
6643:
6639:
6627:
6623:
6619:
6615:
6611:
6607:
6599:
6592:Grothendieck
6590:
6569:
6565:
6561:
6557:
6553:
6549:
6545:
6537:
6533:
6529:
6525:
6521:
6517:
6509:
6508:
6486:
6479:
6474:
6468:
6461:
6442:
6436:constructed
6431:
6420:
6397:
6386:
6381:
6377:
6373:
6371:
6366:
6360:
6257:
6253:
6249:
6241:
6237:
6233:
6229:
6215:
6206:
6204:
6194:
6107:
6103:
6098:
6094:
6086:
6082:
6080:
6075:
6071:
6067:
6063:
6059:
6055:
6050:
6046:
6042:
6038:
6034:
6029:
6025:
6021:
6017:
6013:
6009:
6005:
6001:
5997:
5993:
5992:submanifold
5989:
5985:
5983:
5978:
5976:
5870:
5866:
5864:
5859:
5855:
5853:
5747:
5743:
5739:
5737:
5619:
5615:
5613:
5597:
5535:
5479:
5339:
5335:
5329:
5323:
5319:
5269:
5265:
5263:
5228:by some map
5198:
5088:
5083:
5079:
5043:
5041:
4945:
4900:
4896:
4892:
4831:
4827:
4823:
4819:
4815:
4776:
4772:
4770:
4740:
4735:
4731:
4727:
4723:
4719:
4715:
4707:
4703:
4699:
4695:
4689:
4685:
4681:
4677:
4673:
4668:
4652:
4648:
4644:
4640:
4636:
4628:
4624:
4620:
4616:
4612:
4608:
4604:
4598:
4480:
4476:
4472:
4468:
4464:
4460:
4455:be a closed
4452:
4450:
4313:
4309:
4305:
4301:
4297:
4293:
4289:
4285:
4281:
4277:
4273:
4269:
4265:
4261:
4257:
4253:
4249:
4245:
4243:
4151:
4147:
4143:
4139:
4135:
4131:
4127:
4123:
4119:
4115:
4111:
4107:
4103:
4099:
4095:
4091:
4087:
4083:
4079:
4075:
4071:
4067:
4063:
4059:
4055:
4051:
4047:
4043:
4039:
4035:
4031:
4027:
4024:diagonal map
4021:
4018:The diagonal
4010:
4006:
4002:
3998:
3994:
3990:
3986:
3982:
3978:
3974:
3970:
3966:
3962:
3958:
3950:
3945:
3941:
3937:
3933:
3928:
3924:
3919:
3915:
3911:
3907:
3903:
3899:
3894:
3890:
3886:
3882:
3878:
3874:
3873:= 0 for all
3869:
3865:
3861:
3857:
3852:
3848:
3844:
3840:
3836:
3831:
3827:
3820:
3815:
3811:
3804:
3800:
3796:
3789:
3782:
3778:
3774:
3770:
3766:
3762:
3758:
3754:
3750:
3746:
3736:
3732:
3728:
3724:
3720:
3716:
3712:
3708:
3704:
3700:
3696:
3692:
3688:
3684:
3680:
3676:
3672:
3665:
3661:
3657:
3653:
3649:
3645:
3522:
3514:
3510:
3502:.) Then the
3499:
3495:
3491:
3487:
3483:
3479:
3475:
3471:
3464:
3460:
3456:
3452:
3448:
3442:
3436:
3392:
3388:
3384:
3353:
3349:
3345:
3291:
3287:
3283:
3267:
3263:
3229:
3214:
3211:contractible
3206:
3198:
3196:
3186:
3182:
3178:
3174:
3038:
2931:
2927:
2926:submanifold
2923:
2919:
2915:
2911:
2871:
2867:
2863:
2823:
2821:
2815:
2807:
2743:
2735:
2727:
2715:
2711:
2707:
2699:
2695:
2691:
2687:
2683:
2679:
2671:
2667:
2663:
2659:
2655:
2651:
2647:
2560:transversely
2555:
2551:
2547:
2543:
2539:
2536:
2532:
2528:
2521:
2517:
2513:
2509:
2505:
2501:
2498:
2490:
2485:be a closed
2482:
2474:
2470:
2466:
2458:
2454:
2451:
2447:
2445:
2432:
2309:
2175:
2169:
2078:
2074:
2070:
2066:
2062:
1949:
1946:bilinear map
1941:
1937:
1935:
1929:
1921:
1917:
1913:
1909:
1901:
1893:
1889:
1881:
1827:
1780:vector space
1729:
1395:
1391:
1178:
1174:
1172:open subsets
1167:
1156:
1152:
1141:
1062:
987:
949:
940:
932:
928:
921:
917:
915:
906:
902:
898:
894:
890:
886:
883:coboundaries
882:
878:
874:
870:
866:
862:
858:
854:
853:-cochain on
850:
846:
840:
838:
802:
798:
794:
790:
786:
782:
778:
772:
768:
761:
757:
753:
749:
746:
742:
740:
594:
405:
401:
399:
394:
389:
385:
381:
376:
371:
367:
363:
362:-simplex to
359:
348:
344:
206:
204:
195:
191:
187:
183:
180:homomorphism
175:
171:
167:
156:
155:
137:
132:
128:
123:
118:
114:
110:
102:
98:
94:
90:
68:
43:
29:
10848:Wikiversity
10765:Key results
9950:Dold (1972)
8718:called the
8016:called the
7948:) from the
7442:there is a
6457:cap product
6453:cup product
6449:Eduard Čech
6402:proved the
6363:cup product
5620:cap product
5610:Cap product
5604:Cap product
5322:with group
4691:Euler class
3735:in degree 2
3525:-algebras:
3387:the point (
2818:cohomology.
2526:codimension
2166:graded ring
1951:cup product
1908:, then the
1394:of a space
988:pushforward
837:are called
186:to that of
142:cup product
32:mathematics
10694:CW complex
10635:Continuity
10625:Closed set
10584:cohomology
10453:Thom, René
10203:Birkhäuser
10184:References
9843:Ext groups
9248:for every
9096:Additivity
9088:for every
8671:for every
8575:direct sum
8519:Additivity
8511:for every
7859:See also:
7416:subscheme
6714:Ext groups
6679:Tor groups
6632:left exact
6482:Jean Leray
6416:characters
5328:principal
5272:. Namely,
4463:, and let
3670:hyperplane
3286:in degree
2083:direct sum
1836:CW complex
1776:dual space
1538:Ext groups
1335:There are
1151:maps from
1144:-modules).
413:dual group
397:negative.
105:, for any
44:cohomology
10873:geometric
10868:algebraic
10719:Cobordism
10655:Hausdorff
10650:connected
10567:Geometric
10557:Continuum
10547:Algebraic
10491:120243638
10469:: 17–86,
10376:EMS Press
9998:Thom 1954
9986:Thom 1954
9962:Dold 1972
9922:Citations
9733:∗
9665:∗
9626:∗
9590:∗
9551:∗
9494:∗
9450:∗
9408:∗
9390:cobordism
9355:∗
9351:π
9313:∗
9304:π
9231:α
9218:α
9195:α
9191:∏
9187:→
9070:∩
9041:∗
9033:→
8942:⋯
8939:→
8900:→
8872:∗
8864:→
8836:∗
8828:→
8798:→
8795:⋯
8749:Exactness
8722:(writing
8634:→
8626:α
8613:α
8590:α
8586:⨁
8470:∗
8462:→
8451:∩
8363:⋯
8360:→
8343:−
8330:∂
8327:→
8293:∗
8285:→
8257:∗
8249:→
8225:→
8222:⋯
8171:Exactness
8140:→
8084:→
7818:⊕
7796:≅
7774:⊕
7690:⋯
7687:→
7659:→
7637:→
7596:⊕
7574:→
7552:→
7549:⋯
7514:⟶
7502:↓
7496:↓
7465:⟶
7427:⊂
7401:≥
7336:ℓ
7258:…
7218:…
7185:
7145:ℓ
7131:ℓ
7120:⊗
7104:ℓ
7060:
7050:∈
7041:←
7024:ℓ
6969:≠
6966:ℓ
6944:ℓ
6843:, called
6780:), where
6480:In 1946,
6462:In 1944,
6432:In 1936,
6398:In 1934,
6387:In 1931,
6329:−
6312:→
6290:×
6161:−
6148:≅
6145:→
5981:a field.
5922:
5909:→
5822:∗
5814:→
5794:∗
5786:×
5766:∗
5703:−
5692:→
5664:×
5633:∩
5549:∈
5447:π
5402:π
5395:
5239:→
4998:≅
4993:→
4914:→
4581:≅
4553:→
4530:−
4519:×
4457:connected
4411:×
4386:∈
4369:∗
4344:∗
4327:×
4196:∈
4187:×
4176:∗
4172:Δ
3607:∗
3593:⊗
3572:∗
3564:≅
3549:×
3538:∗
3144:∗
3102:−
3094:−
3054:∗
3011:−
2967:∈
2947:∗
2606:∈
2597:∩
2402:∗
2394:→
2374:∗
2361:∗
2327:→
2321::
2268:∈
2234:∈
2202:−
2121:⨁
2097:∗
2015:→
1987:×
1928:for each
1886:dimension
1710:→
1668:
1653:→
1625:→
1596:
1588:−
1574:
1554:→
1518:⋯
1515:→
1481:→
1459:→
1437:→
1409:→
1406:⋯
1321:⋯
1318:→
1290:→
1281:∩
1262:→
1240:⊕
1218:→
1196:→
1193:⋯
1149:homotopic
1105:→
1078:∗
1030:→
1003:∗
971:→
873:) and im(
843:-cochains
839:singular
823:∗
727:⋯
724:←
719:∗
711:−
691:−
678:←
669:∗
637:←
628:∗
609:←
606:⋯
576:∗
563:→
558:∗
550:−
531:−
494:∂
432:∗
374:"), and ∂
327:⋯
324:→
316:−
292:∂
286:→
253:∂
247:→
224:→
221:⋯
87:pullbacks
60:functions
10900:Category
10838:Wikibook
10816:Category
10704:Manifold
10672:Homotopy
10630:Interior
10621:Open set
10579:Homology
10528:Topology
10455:(1954),
10387:(1999),
10333:(2001),
10300:(1977),
10267:(1952),
10230:(1972),
10193:(1989),
10133:May 1999
10097:May 1999
9910:See also
9533:K-theory
9258:spectrum
8960:Excision
8743:Homotopy
8382:Excision
8053:Homotopy
7950:category
7934:functors
7328:and the
6487:In 1948
6224:founded
6209:, which
5438:, where
5332:-bundles
4676:of rank
4252:, write
3927:for all
3757:), with
3660:), with
3282:), with
3193:Examples
2487:oriented
2437:algebras
2347:pullback
2178:. It is
1912:-module
1832:manifold
1389:subspace
1387:for any
1063:pullback
945:integers
924:to be a
879:cocycles
113: :
75:geometry
71:topology
10863:general
10665:uniform
10645:compact
10596:Digital
10483:0061823
10446:0385836
10416:1702278
10378:, 2001
10359:1867354
10324:0463157
10291:0050886
10254:0415602
10221:0995842
10177:, 7.68.
9292:Stable
8779:,∅) → (
8204:,∅) → (
7960:,
5474:is the
4712:section
4633:torsion
3898:= 0 if
3719:=1,...,
3445:-module
3340:is the
3270:) (the
3185:inside
2814:on mod
2562:, then
2493:. Then
2164:into a
1898:compact
1778:of the
1339:groups
937:modules
512:by its
411:by its
380:is the
354:is the
107:mapping
79:algebra
10858:Topics
10660:metric
10535:Fields
10489:
10481:
10444:
10434:
10414:
10404:
10357:
10347:
10322:
10312:
10289:
10279:
10252:
10242:
10219:
10209:
9098:: If (
8730:) for
8521:: If (
8020:(here
7964:) (so
7952:of CW-
6811:smooth
6677:, the
6471:axioms
6423:Moscow
6089:has a
5618:, the
5338:. For
4814:whose
4753:, and
4688:, the
4619:) and
4316:) is:
4300:) and
4138:) and
4074:) and
4009:(with
3997:(with
3989:/2 or
3906:, and
3723:. The
3383:, and
3234:sphere
2776:or in
2670:, and
745:, the
656:
305:
64:chains
10640:Space
10487:S2CID
10394:(PDF)
9262:Adams
9142:) → (
8990:) → (
8962:: If
8565:) → (
8413:) → (
8385:: If
8055:: If
7954:pairs
7904:: if
6514:sheaf
5334:over
4599:is a
3826:,...,
3810:,...,
3794:genus
3648:with
3517:is a
3441:free
3344:over
3299:torus
3280:ideal
3274:of a
2914:from
2738:with
2710:. If
2678:, if
2477:is).
1924:) is
1896:is a
1892:. If
1834:or a
1727:field
889:)/im(
767:)/im(
10432:ISBN
10402:ISBN
10345:ISBN
10310:ISBN
10277:ISBN
10240:ISBN
10207:ISBN
8970:and
8771:and
8704:) →
8393:and
8195:and
8000:) →
7885:for
7182:Proj
6958:for
6576:and
6447:and
6220:and
6066:and
5865:For
5742:and
4698:) ∈
4647:) ≅
4268:and
4248:and
4094:(or
4054:and
3877:and
3819:and
3715:for
3656:/2/(
3478:and
2740:real
2720:Thom
2682:and
2554:and
2546:and
2481:Let
2345:the
2065:and
1532:The
1177:and
1162:The
1147:Two
893:) =
881:and
485:and
146:ring
93:and
77:and
38:and
10471:doi
9720:if
8978:: (
8787:):
8775:: (
8401:: (
8200:: (
7873:or
7038:lim
6851:in
6744:).
6716:Ext
6708:of
6681:Tor
6650:).
6614:on
6544:on
6236:an
6093:in
6020:of
5996:of
5913:Hom
5478:of
5392:Hom
5318:of
5082:to
4834:of
4714:of
4042:↦ (
4026:Δ:
3985:is
3949:= −
3792:of
3781:in
3769:in
3749:is
3739:+1.
3675:in
3521:of
3463:= −
3348:on
3262:is
3037:of
2930:of
2690:or
2531:in
2524:of
2465:of
2174:of
1888:of
1826:If
1657:Hom
1558:Ext
1155:to
943:of
865:to
752:of
343:of
194:to
136:on
101:on
30:In
10902::
10485:,
10479:MR
10477:,
10467:28
10465:,
10459:,
10442:MR
10440:,
10430:,
10426:,
10412:MR
10410:,
10400:,
10396:,
10374:,
10368:,
10355:MR
10353:,
10343:,
10339:,
10320:MR
10318:,
10308:,
10287:MR
10285:,
10275:,
10271:,
10263:;
10250:MR
10248:,
10238:,
10234:,
10217:MR
10215:,
10205:,
10201:,
10160:.
10029:^
9694:,
9277:.
9256:A
9154::
8767:→
8763::
8696::
8577::
8214::
8190:→
8186::
8042:−1
8028:−1
8008:−1
7996:,
7983::
7889:,
7869:,
7032::=
6797:.
6776:,
6755:,
6642:↦
6622:,
6588:.
6495:.
6425:,
6418:.
6384:.
6369:.
6197:.
6078:.
6074:−
5869:=
5862:.
5600:.
5261:.
5086:.
4757:.
4749:,
4738:.
4694:χ(
4655:.
4304:∈
4288:∈
4280:→
4276:×
4272::
4264:→
4260:×
4256::
4142:∈
4126:∈
4114:×
4106:∈
4102:×
4078:∈
4062:∈
4038:,
4034:×
4030:→
4007:RP
4005:×
4003:RP
3995:RP
3923:=
3902:≠
3881:,
3856:=
3783:CP
3779:CP
3771:CP
3767:CP
3753:/(
3747:CP
3729:RP
3697:RP
3681:RP
3677:RP
3673:RP
3646:RP
3513:×
3465:xy
3461:yx
3457:xy
3266:/(
3189:.
2730:,
2666:,
2658:+
2650:∩
2504:≅
2079:uv
2073:∪
1954::
1732:,
1713:0.
1185::
947:.
901:,
797:,
785:,
775:−1
437::=
213::
174:→
170::
131:∘
117:→
42:,
10520:e
10513:t
10506:v
10473::
10380:.
10164:.
9770:∞
9767:E
9758:X
9744:)
9741:X
9738:(
9729:E
9714:E
9705:.
9676:)
9673:X
9670:(
9661:u
9657:k
9637:)
9634:X
9631:(
9622:K
9601:)
9598:X
9595:(
9586:o
9582:k
9562:)
9559:X
9556:(
9547:O
9543:K
9528:.
9508:,
9505:)
9502:X
9499:(
9490:U
9486:M
9464:,
9461:)
9458:X
9455:(
9446:O
9442:S
9439:M
9419:)
9416:X
9413:(
9404:O
9400:M
9374:.
9371:)
9368:X
9365:(
9360:S
9327:.
9324:)
9321:X
9318:(
9308:S
9252:.
9250:i
9236:)
9227:A
9223:,
9214:X
9210:(
9205:i
9201:h
9184:)
9181:A
9178:,
9175:X
9172:(
9167:i
9163:h
9148:A
9146:,
9144:X
9139:α
9135:A
9133:,
9130:α
9126:X
9121:α
9117:A
9115:,
9112:α
9108:X
9104:A
9102:,
9100:X
9092:.
9090:i
9076:)
9073:B
9067:A
9064:,
9061:A
9058:(
9053:i
9049:h
9037:f
9028:)
9025:B
9022:,
9019:X
9016:(
9011:i
9007:h
8996:B
8994:,
8992:X
8988:B
8986:∩
8984:A
8982:,
8980:A
8976:f
8972:B
8968:A
8964:X
8945:.
8936:)
8933:A
8930:,
8927:X
8924:(
8919:1
8916:+
8913:i
8909:h
8903:d
8895:)
8892:A
8889:(
8884:i
8880:h
8868:f
8859:)
8856:X
8853:(
8848:i
8844:h
8832:g
8823:)
8820:A
8817:,
8814:X
8811:(
8806:i
8802:h
8785:A
8783:,
8781:X
8777:X
8773:g
8769:X
8765:A
8761:f
8757:A
8755:,
8753:X
8736:A
8734:(
8732:h
8728:A
8726:(
8724:h
8716:)
8714:A
8712:,
8710:X
8708:(
8706:h
8702:A
8700:(
8698:h
8694:d
8689:i
8685:h
8675:.
8673:i
8659:)
8656:A
8653:,
8650:X
8647:(
8642:i
8638:h
8631:)
8622:A
8618:,
8609:X
8605:(
8600:i
8596:h
8571:A
8569:,
8567:X
8562:α
8558:A
8556:,
8553:α
8549:X
8544:α
8540:A
8538:,
8535:α
8531:X
8527:A
8525:,
8523:X
8515:.
8513:i
8499:)
8496:B
8493:,
8490:X
8487:(
8482:i
8478:h
8466:f
8457:)
8454:B
8448:A
8445:,
8442:A
8439:(
8434:i
8430:h
8419:B
8417:,
8415:X
8411:B
8409:∩
8407:A
8405:,
8403:A
8399:f
8395:B
8391:A
8387:X
8366:.
8357:)
8354:A
8351:(
8346:1
8340:i
8336:h
8322:)
8319:A
8316:,
8313:X
8310:(
8305:i
8301:h
8289:g
8280:)
8277:X
8274:(
8269:i
8265:h
8253:f
8244:)
8241:A
8238:(
8233:i
8229:h
8212:)
8210:A
8208:,
8206:X
8202:X
8198:g
8192:X
8188:A
8184:f
8179:A
8177:,
8175:X
8155:)
8152:B
8149:,
8146:Y
8143:(
8137:)
8134:A
8131:,
8128:X
8125:(
8122::
8119:g
8099:)
8096:B
8093:,
8090:Y
8087:(
8081:)
8078:A
8075:,
8072:X
8069:(
8066::
8063:f
8046:A
8044:(
8040:i
8036:h
8032:A
8030:(
8026:i
8022:h
8014:)
8012:A
8010:(
8006:i
8002:h
7998:A
7994:X
7992:(
7989:i
7985:h
7980:i
7977:∂
7970:A
7966:X
7962:A
7958:X
7956:(
7946:i
7941:i
7937:h
7918:i
7914:P
7912:(
7910:H
7906:P
7837:)
7834:E
7831:(
7826:n
7822:H
7815:)
7812:X
7809:(
7804:n
7800:H
7793:)
7790:Z
7787:(
7782:n
7778:H
7771:)
7768:)
7765:X
7762:(
7757:Z
7753:l
7749:B
7746:(
7741:n
7737:H
7713:Z
7684:)
7681:X
7678:(
7673:1
7670:+
7667:n
7663:H
7656:)
7653:E
7650:(
7645:n
7641:H
7634:)
7631:)
7628:X
7625:(
7620:Z
7616:l
7612:B
7609:(
7604:n
7600:H
7593:)
7590:Z
7587:(
7582:n
7578:H
7571:)
7568:X
7565:(
7560:n
7556:H
7519:X
7509:Z
7489:)
7486:X
7483:(
7478:Z
7474:l
7470:B
7460:E
7430:X
7424:Z
7404:2
7374:)
7369:q
7364:F
7359:(
7356:X
7316:)
7312:C
7308:(
7305:X
7281:)
7275:)
7269:k
7265:f
7261:,
7255:,
7250:1
7246:f
7241:(
7235:]
7229:n
7225:x
7221:,
7215:,
7210:0
7206:x
7201:[
7196:Z
7189:(
7179:=
7176:X
7150:.
7140:Q
7126:Z
7116:)
7113:)
7108:n
7100:(
7096:/
7091:Z
7087:;
7084:X
7081:(
7076:k
7071:t
7068:e
7064:H
7054:N
7047:n
7029:)
7019:Q
7014:;
7011:X
7008:(
7003:k
6999:H
6972:p
6923:p
6895:p
6866:n
6861:P
6827:0
6795:X
6791:Z
6786:X
6782:Z
6778:E
6773:X
6769:Z
6765:E
6763:,
6761:X
6759:(
6757:H
6753:X
6749:E
6742:N
6740:,
6738:M
6736:(
6733:R
6728:N
6726:,
6724:M
6722:(
6719:R
6710:R
6706:N
6702:R
6699:⊗
6697:M
6693:N
6691:,
6689:M
6687:(
6684:i
6675:R
6667:X
6655:X
6648:X
6646:(
6644:E
6640:E
6628:X
6626:(
6624:E
6620:X
6616:X
6612:E
6608:X
6600:X
6570:X
6566:X
6562:A
6560:,
6558:X
6556:(
6554:H
6550:A
6546:X
6538:i
6534:E
6532:,
6530:X
6528:(
6526:H
6522:X
6518:E
6382:X
6378:X
6374:i
6367:M
6346:,
6343:)
6340:M
6337:(
6332:n
6326:j
6323:+
6320:i
6316:H
6309:)
6306:M
6303:(
6298:j
6294:H
6287:)
6284:M
6281:(
6276:i
6272:H
6258:n
6254:j
6250:i
6242:j
6238:i
6234:M
6230:n
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