10766:
10549:
10787:
10755:
10824:
10797:
10777:
3222:
1723:
7867:
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
7824:, 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.
9227:
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
9216:) 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
8319:
737:
7820:). (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
7107:
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
9211:
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
8898:
7476:
6882:, 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 Grothendieck topology and used sheaf cohomology over the
1546:
337:
7643:
7238:
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
8160:
601:
8733:
1528:
9207:
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.
8612:
9189:
7790:
6958:
3637:
7394:
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
9029:
8452:
4594:
5733:
6805:
There are numerous machines built for computing the cohomology of algebraic varieties. The simplest case being the determination of cohomology for smooth projective varieties over a field of characteristic
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:
5532:
5195:
7330:
6830:
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 hypersurface in
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
7487:
5124:
9327:
9280:
8108:
8052:
7118:
6857:
2644:
7272:
4889:
2343:
1401:
9223:
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:
9417:
6950:
5472:
4942:
3433:
3338:
9461:
7386:
5382:
5312:
5259:
1878:
1821:
1772:
9629:
9554:
9515:
9372:
3035:
835:
9697:
9590:
2908:
2860:
984:
8523:
3171:
3081:
7360:
4812:
7292:
6924:
5226:
5151:
3381:
3260:
9100:
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
10827:
7666:
6904:
6880:
6824:
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:
8944:
8367:
8314:{\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 }
82:
6113:
7102:{\displaystyle H^{k}(X;\mathbb {Q} _{\ell }):=\varprojlim H_{et}^{k}(X;\mathbb {Z} /(\ell ^{n}))\otimes _{\mathbb {Z} _{\ell }}\mathbb {Q} _{\ell }}
7674:
1957:
8893:{\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
7471:{\displaystyle {\begin{matrix}E&\longrightarrow &Bl_{Z}(X)\\\downarrow &&\downarrow \\Z&\longrightarrow &X\end{matrix}}}
4157:
10186:
7336:
which behave similarly to singular cohomology. There is a conjectured theory of motives which underlie all of the Weil cohomology theories.
9725:
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
6886:
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:
7868:
space, not directly accessible from singular cohomology. (In this context, singular cohomology is often called "ordinary cohomology".)
5628:
518:
10461:
9858:
7332:
whenever the variety is smooth over both fields. In addition to these cohomology theories there are other cohomology theories called
10815:
10810:
10404:
10378:
10348:
10291:
10256:
10223:
10153:
7813:
6492:
9203:
determines both a generalized homology theory and a generalized cohomology theory. A fundamental result by Brown, Whitehead, and
86:
9735:
10805:
9708:
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:
347:
is the homology of this chain complex (the kernel of one homomorphism modulo the image of the previous one). In more detail,
9634:
7821:
6470:
10707:
10318:
4954:
2810:
to make the product graded-commutative on the nose. The failure of graded-commutativity at the cochain level leads to the
1533:
8622:
The axioms for a generalized cohomology theory are obtained by reversing the arrows, roughly speaking. In more detail, a
5387:
417:
10848:
2559:
6661:(but usually now just "cohomology"). From that point of view, sheaf cohomology becomes a sequence of functors from the
4766:
1163:
10340:
10313:
7803:
2565:
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:
10715:
10283:
10215:
9463:
and so on. Complex cobordism has turned out to be especially powerful in homotopy theory. It is closely related to
6217:
3518:
63:
5541:
4750:
9755:
9224:
1925:
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
10786:
10514:
9094:
3743:
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
5485:
2718:. In both cases, the cup product can again be described in terms of intersections of submanifolds. Note that
10800:
9815:
6730:) can be viewed as a "cohomology theory" in each variable, the right derived functors of the Hom functor Hom
6415:
2754:. In fact, it is impossible to modify the definition of singular cochains with coefficients in the integers
160:
7825:
7638:{\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:
10735:
10730:
10656:
10533:
10521:
10494:
10454:
9840:
9825:
9775:
9760:
9283:
8634:) from the category of CW-pairs to the category of abelian groups, together with a natural transformation
7916:
7892:
7333:
6591:
4711:
210:
59:
7297:
7233:{\displaystyle X={\text{Proj}}\left({\frac {\mathbb {Z} \left}{\left(f_{1},\ldots ,f_{k}\right)}}\right)}
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 }
10577:
10504:
9810:
9740:
9217:
8517:
6466:
overcame the technical limitations, and gave the modern definition of singular homology and cohomology.
3642:
2346:
1936:
On the other hand, cohomology has a crucial structure that homology does not: for any topological space
1137:
10765:
5092:
9288:
9241:
8057:
8001:
6833:
10725:
10677:
10651:
10499:
9652:
Many of these theories carry richer information than ordinary cohomology, but are harder to compute.
9335:
groups, based on studying a space by considering all maps from it to manifolds: unoriented cobordism
7246:
6585:
4837:
4742:
4664:
2731:
2313:
1541:
936:
2779:
10776:
10572:
9780:
9645:
9200:
7864:
6595:
6513:
6225:
4446:
3793:
3279:
2494:
2440:
2436:
1900:
manifold (possibly with boundary), or a CW complex with finitely many cells in each dimension, and
1726:
1182:
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:
10770:
10720:
10641:
10631:
10509:
10489:
10429:
9845:
9835:
9770:
9730:
9377:
7837:
7829:
7808:
There are various ways to define cohomology for topological spaces (such as singular cohomology,
6929:
6631:
6573:
6403:
6392:
5441:
4906:
3398:
3303:
2811:
2179:
355:
145:
39:
10740:
10308:
9750:
9424:
7809:
7365:
6634:, but not necessarily right exact. Grothendieck defined sheaf cohomology groups to be the right
6437:
6429:
and
Alexander both introduced cohomology and tried to construct a cohomology product structure.
5345:
5275:
5231:
3503:
1841:
1784:
1735:
10331:
9595:
9520:
9481:
9338:
3001:
808:
148:
structure. Because of this feature, cohomology is usually a stronger invariant than homology.
10758:
10624:
10582:
10447:
10374:
10344:
10287:
10252:
10219:
10182:
10149:
10133:
9820:
9800:
9765:
9666:
9559:
9419:
7833:
6883:
6426:
6411:
6407:
6221:
6210:
6090:
5475:
5264:
There is a related description of the first cohomology with coefficients in any abelian group
2877:
2829:
2747:
1388:
1336:
957:
910:
340:
179:
106:
51:
6695:) form a "homology theory" in each variable, the left derived functors of the tensor product
3134:
3044:
10538:
10484:
10413:
10240:
10203:
10138:
9830:
9805:
9236:
8324:
7896:
7817:
7342:
6662:
6603:
6581:
6577:
6504:
6463:
6245:
5327:
4782:
4754:
4632:
3341:
925:
78:
10425:
10388:
10358:
10301:
10266:
10233:
10196:
10163:
7277:
6909:
5204:
5129:
4244:
Alternatively, the external product can be defined in terms of the cup product. For spaces
3359:
3238:
10597:
10592:
10421:
10384:
10370:
10354:
10297:
10262:
10248:
10229:
10207:
10192:
10178:
10159:
9790:
9745:
9638:
6827:
6658:
6635:
6444:
6433:
6388:
4600:
4456:
3275:
2675:
2170:
1905:
596:
55:
35:
4631:) have the same (finite) dimension. Likewise, the product on integral cohomology modulo
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
1523:{\displaystyle \cdots \to H^{i}(X,Y)\to H^{i}(X)\to H^{i}(Y)\to H^{i+1}(X,Y)\to \cdots }
10687:
10619:
9795:
9468:
7651:
6889:
6865:
6809:
6541:
6448:
6399:
5315:
2751:
2061:
defined by an explicit formula on singular cochains. The product of cohomology classes
199:
164:
10277:
5049:
10842:
10697:
10607:
10587:
10433:
10273:
10170:
10145:
9704:
6488:
6410:. This (in rather special cases) provided an interpretation of Poincaré duality and
5988:
be an oriented manifold, not necessarily compact. Then a closed oriented codimension-
4670:
3507:
3271:
2703:
2486:
2462:
1897:
595:
This has the effect of "reversing all the arrows" of the original complex, leaving a
47:
10790:
3957:
On any topological space, graded-commutativity of the cohomology ring implies that 2
10682:
10602:
10548:
10327:
9464:
4023:
3210:
1945:
1779:
10395:
5596:
corresponding to the trivial double covering, the disjoint union of two copies of
2719:
6747:
Sheaf cohomology can be identified with a type of Ext group. Namely, for a sheaf
202:, the cohomology ring tends to be computable in practice for spaces of interest.
10780:
10692:
9213:
9204:
8607:{\displaystyle \bigoplus _{\alpha }h_{i}(X_{\alpha },A_{\alpha })\to h_{i}(X,A)}
6678:
6473:
defining a homology or cohomology theory, discussed below. In their 1952 book,
6456:
6452:
6362:
5609:
4746:
4690:
3440:
2739:
2525:
2165:
1950:
1171:
141:
31:
17:
10636:
10567:
7339:
Another useful computational tool is the blowup sequence. Given a codimension
6481:
5314:
is in one-to-one correspondence with the set of isomorphism classes of Galois
3669:
2082:
1835:
1775:
513:
412:
6862:
When considering varieties over a finite field, or a field of characteristic
6372:
Alexander had by 1930 defined a first notion of a cochain, by thinking of an
3209:
in degree 0. By homotopy invariance, this is also the cohomology ring of any
10661:
9785:
9332:
9184:{\displaystyle h^{i}(X,A)\to \prod _{\alpha }h^{i}(X_{\alpha },A_{\alpha })}
7785:{\displaystyle H^{n}(Bl_{Z}(X))\oplus H^{n}(Z)\cong H^{n}(X)\oplus H^{n}(E)}
6713:
1885:
1537:
1398:. They are related to the usual cohomology groups by a long exact sequence:
1148:
2694:
to make the intersection transverse. More generally, without assuming that
9703:. 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
3221:
10646:
10614:
10563:
10470:
9475:
4710:). Informally, the Euler class is the class of the zero set of a general
3632:{\displaystyle H^{*}(X\times Y,R)\cong H^{*}(X,R)\otimes _{R}H^{*}(Y,R).}
3233:
1831:
74:
70:
6594:
elegantly defined and characterized sheaf cohomology in the language of
6572:). Starting in the 1950s, sheaf cohomology has become a central part of
10417:
9478:, based on studying a space by considering all vector bundles over it:
7876:
4431:{\displaystyle u\times v=(f^{*}(u))(g^{*}(v))\in H^{i+j}(X\times Y,R).}
3197:
In what follows, cohomology is taken with coefficients in the integers
944:
9717:, 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).}
10247:, Graduate Texts in Mathematics, vol. 52, New York, Heidelberg:
6422:
5967:{\displaystyle H^{i}(X,R)\to \operatorname {Hom} _{R}(H_{i}(X,R),R),}
10100:
7915:
is a subcomplex) to the category of abelian groups, together with a
7843:
One of the
Eilenberg–Steenrod axioms for a cohomology theory is the
7243:
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
9641:, Morava E-theory, and other theories built from complex cobordism.
5126:(defined up to homotopy equivalence) can be taken to be the circle
9024:{\displaystyle h^{i}(X,B){\overset {f_{*}}{\to }}h^{i}(A,A\cap B)}
8702:) induces a long exact sequence in cohomology, via the inclusions
8447:{\displaystyle h_{i}(A,A\cap B){\overset {f_{*}}{\to }}h_{i}(X,B)}
4589:{\displaystyle H^{i}(X,F)\times H^{n-i}(X,F)\to H^{n}(X,F)\cong F}
3298:
3220:
9975:
9973:
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).
8124:) induces a long exact sequence in homology, via the inclusions
7668:
is smooth, then the connecting morphisms are all trivial, hence
6395:. This result can be stated more simply in terms of cohomology.
10443:
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
10399:
4284:
for the two projections. Then the external product of classes
4745:
for vector bundles that take values in cohomology, including
3707:
spanned by the element 1 in degree 0 together with copies of
140:. The most important cohomology theories have a product, the
8688:: 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
10439:
10400:"Quelques propriétés globales des variétés différentiables"
10090:, p. 117, 331, Theorem 9.27; Corollary 14.36; Remarks.
6859:
can be determined from the degree of the polynomial alone.
857:
can be identified with a function from the set of singular
8110:, 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
3083:
restricts to zero in the cohomology of the open subset
2450:
means a compact manifold (without boundary), whereas a
9282:
The corresponding homology theory is used more often:
9231:
Some examples of generalized cohomology theories are:
7399:
5153:. So the description above says that every element of
4826:. This space has the remarkable property that it is a
10101:"Are spectra really the same as cohomology theories?"
9669:
9598:
9562:
9523:
9484:
9427:
9380:
9341:
9291:
9244:
9103:
8947:
8736:
8526:
8370:
8163:
8060:
8004:
7677:
7654:
7490:
7481:
From this there is an associated long exact sequence
7397:
7368:
7345:
7300:
7280:
7249:
7121:
6961:
6932:
6912:
6892:
6868:
6836:
6812:
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:
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
10706:
10670:
10556:
10477:
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
10137:
9699:has the structure of a graded ring for each space
9691:
9623:
9584:
9548:
9509:
9455:
9411:
9366:
9321:
9274:
9183:
9023:
8892:
8606:
8446:
8313:
8102:
8046:
7784:
7660:
7637:
7470:
7380:
7354:
7324:
7286:
7266:
7232:
7101:
6944:
6918:
6898:
6874:
6851:
6818:
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:
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}^{*}.}
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
10140:History of Algebraic and Differential Topology
9895:, 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
10455:
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
8:
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
9049:) is the disjoint union of a set of pairs (
8472:) 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
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
10823:
10796:
10462:
10448:
10440:
10367:Algebraic Topology — Homology and Homotopy
10063:
7796:Axioms and generalized cohomology theories
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)}
10051:
9674:
9668:
9631:(complex connective K-theory), and so on.
9606:
9597:
9567:
9561:
9531:
9522:
9492:
9483:
9474:Various different flavors of topological
9435:
9426:
9391:
9379:
9349:
9340:
9301:
9296:
9290:
9254:
9249:
9243:
9172:
9159:
9146:
9136:
9108:
9102:
8994:
8982:
8973:
8952:
8946:
8854:
8840:
8825:
8813:
8804:
8789:
8777:
8768:
8747:
8735:
8583:
8567:
8554:
8541:
8531:
8525:
8423:
8411:
8402:
8375:
8369:
8281:
8267:
8246:
8234:
8225:
8210:
8198:
8189:
8174:
8162:
8059:
8003:
7767:
7745:
7723:
7698:
7682:
7676:
7653:
7608:
7586:
7561:
7545:
7523:
7501:
7489:
7419:
7398:
7396:
7367:
7344:
7313:
7309:
7308:
7299:
7279:
7257:
7256:
7248:
7213:
7194:
7173:
7154:
7141:
7140:
7137:
7128:
7120:
7093:
7089:
7088:
7079:
7075:
7074:
7072:
7056:
7044:
7040:
7039:
7024:
7016:
6999:
6987:
6983:
6982:
6966:
6960:
6931:
6911:
6891:
6867:
6843:
6839:
6838:
6835:
6811:
6318:
6296:
6274:
6268:
6156:
6142:
6121:
6115:
5931:
5915:
5887:
5881:
5820:
5792:
5764:
5758:
5698:
5670:
5642:
5630:
5575:
5571:
5570:
5555:
5543:
5534:classifies the double covering spaces of
5513:
5509:
5508:
5493:
5487:
5449:
5443:
5404:
5389:
5353:
5347:
5283:
5277:
5245:
5233:
5212:
5206:
5180:
5179:
5164:
5158:
5137:
5131:
5103:
5102:
5094:
5051:
5008:
4996:
4991:
4989:
4988:
4956:
4944:. More precisely, pulling back the class
4908:
4845:
4839:
4784:
4726:, since then a general smooth section of
4559:
4525:
4497:
4491:
4392:
4367:
4342:
4321:
4202:
4174:
4159:
3605:
3595:
3570:
3536:
3530:
3419:
3409:
3400:
3367:
3361:
3352:generators in degree 1. For example, let
3324:
3314:
3305:
3290:. In terms of Poincaré duality as above,
3246:
3240:
3142:
3136:
3100:
3088:
3052:
3046:
3009:
3003:
2973:
2945:
2939:
2885:
2879:
2837:
2831:
2788:
2784:
2783:
2781:
2762:
2761:
2759:
2612:
2567:
2400:
2372:
2359:
2353:
2315:
2274:
2240:
2211:
2187:
2133:
2123:
2095:
2089:
2021:
1993:
1965:
1959:
1849:
1843:
1792:
1786:
1743:
1737:
1693:
1692:
1677:
1661:
1660:
1659:
1631:
1608:
1607:
1583:
1567:
1562:
1561:
1560:
1548:
1487:
1465:
1443:
1415:
1403:
1350:
1344:
1296:
1268:
1246:
1224:
1202:
1190:
1111:
1089:
1076:
1070:
1036:
1014:
1001:
995:
959:
821:
816:
810:
717:
706:
686:
681:
676:
674:
673:
667:
662:
645:
640:
635:
633:
632:
626:
615:
603:
574:
569:
556:
545:
526:
520:
496:
490:
457:
439:
430:
425:
419:
311:
294:
289:
284:
282:
281:
275:
255:
250:
245:
243:
242:
230:
218:
8626:is a sequence of contravariant functors
6564:) coincide with singular cohomology for
3447:of the form: the element 1 in degree 0,
198:. Unlike more subtle invariants such as
10117:
10087:
10027:
10015:
10003:
9991:
9979:
9964:
9952:
9916:
9888:
9876:
9869:
5527:{\displaystyle H^{1}(X,\mathbb {Z} /2)}
4891:, and every cohomology class of degree
916:In what follows, the coefficient group
10333:A Concise Course in Algebraic Topology
9228: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
89:in geometric situations: given spaces
5190:{\displaystyle H^{1}(X,\mathbb {Z} )}
2734:says that the singular cohomology of
7:
9940:
9928:
9904:
9892:
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
10075:
10039:
7325:{\displaystyle X(\mathbb {F} _{q})}
7112:If we have a scheme of finite type
4607:. In particular, the vector spaces
2991:{\displaystyle f^{*}()\in H^{i}(X)}
9907:, Propositions IV.8.12 and V.4.11.
9859:complex-oriented cohomology theory
8272:
6826:. Tools from Hodge theory, called
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:
10405:Commentarii Mathematici Helvetici
10212:Foundations of Algebraic Topology
8516:) 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
10822:
10795:
10785:
10775:
10764:
10754:
10753:
10547:
9322:{\displaystyle \pi _{*}^{S}(X).}
9275:{\displaystyle \pi _{S}^{*}(X).}
8103:{\displaystyle g:(X,A)\to (Y,B)}
8047:{\displaystyle f:(X,A)\to (Y,B)}
6852:{\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
9093:) induce an isomorphism to the
7267:{\displaystyle X(\mathbb {C} )}
5854:makes the singular homology of
4884:{\displaystyle H^{j}(K(A,j),A)}
4459:oriented manifold of dimension
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:
10175:Lectures on Algebraic Topology
9686:
9680:
9618:
9612:
9579:
9573:
9543:
9537:
9504:
9498:
9447:
9441:
9403:
9397:
9361:
9355:
9313:
9307:
9266:
9260:
9178:
9152:
9129:
9126:
9114:
9018:
9000:
8975:
8970:
8958:
8881:
8878:
8866:
8842:
8837:
8831:
8806:
8801:
8795:
8770:
8765:
8753:
8740:
8601:
8589:
8576:
8573:
8547:
8441:
8429:
8404:
8399:
8381:
8302:
8299:
8293:
8269:
8264:
8252:
8227:
8222:
8216:
8191:
8186:
8180:
8167:
8097:
8085:
8082:
8079:
8067:
8041:
8029:
8026:
8023:
8011:
7779:
7773:
7757:
7751:
7735:
7729:
7713:
7710:
7704:
7688:
7629:
7626:
7620:
7601:
7598:
7592:
7579:
7576:
7573:
7567:
7551:
7535:
7529:
7516:
7513:
7507:
7494:
7456:
7444:
7438:
7431:
7425:
7407:
7319:
7304:
7261:
7253:
7065:
7062:
7049:
7030:
6993:
6972:
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:
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:
1:
9592:(complex periodic K-theory),
9331:Various different flavors of
8909:is the union of subcomplexes
8624:generalized cohomology theory
8332: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}}
9556:(real connective K-theory),
7814:Alexander–Spanier cohomology
7388:there is a Cartesian square
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
10341:University of Chicago Press
10314:Encyclopedia of Mathematics
9412:{\displaystyle MSO^{*}(X),}
7873:generalized homology theory
7804:List of cohomology theories
6945:{\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
2702:with an orientation on its
10865:
10716:Banach fixed-point theorem
10284:Cambridge University Press
10216:Princeton University Press
9517:(real periodic K-theory),
9456:{\displaystyle MU^{*}(X),}
7801:
7381:{\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
10749:
10545:
9756:Coherent sheaf cohomology
9721:Other cohomology theories
9635:Brown–Peterson cohomology
9624:{\displaystyle ku^{*}(X)}
9549:{\displaystyle ko^{*}(X)}
9510:{\displaystyle KO^{*}(X)}
9367:{\displaystyle MO^{*}(X)}
9225:weak homotopy equivalence
8941:) induces an isomorphism
8364:) induces an isomorphism
7822:Eilenberg–Steenrod axioms
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
10365:Switzer, Robert (1975),
9736:André–Quillen cohomology
9692:{\displaystyle E^{*}(X)}
9585:{\displaystyle K^{*}(X)}
9067:), then the inclusions (
8490:), then the inclusions (
7851:is a single point, then
7334: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
9816: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
10771:Mathematics portal
10671:Metrics and properties
10657:Second-countable space
9841:Non-abelian cohomology
9826:Lie algebra cohomology
9776:Equivariant cohomology
9761:Crystalline cohomology
9693:
9625:
9586:
9550:
9511:
9457:
9413:
9368:
9323:
9284:stable homotopy groups
9276:
9185:
9025:
8894:
8681:,∅)). The axioms are:
8608:
8448:
8315:
8104:
8048:
7991:,∅)). The axioms are:
7917:natural transformation
7840:for smooth manifolds.
7836:for CW complexes, and
7786:
7662:
7639:
7472:
7382:
7356:
7355:{\displaystyle \geq 2}
7326:
7288:
7268:
7234:
7103:
6946:
6920:
6900:
6876:
6853:
6820:
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
9811:Hochschild cohomology
9694:
9626:
9587:
9551:
9512:
9458:
9414:
9369:
9324:
9277:
9186:
9026:
8917:, then the inclusion
8895:
8663:boundary homomorphism
8609:
8449:
8340:, then the inclusion
8316:
8105:
8049:
7977:) is a shorthand for
7961:boundary homomorphism
7826:simplicial cohomology
7787:
7663:
7640:
7473:
7383:
7357:
7327:
7289:
7287:{\displaystyle \ell }
7269:
7235:
7104:
6952:. This is defined as
6947:
6926:-adic cohomology for
6921:
6919:{\displaystyle \ell }
6901:
6877:
6854:
6821:
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
10726:Invariance of domain
10678:Euler characteristic
10652:Bundle (mathematics)
9667:
9655:A cohomology theory
9596:
9560:
9521:
9482:
9425:
9378:
9339:
9289:
9242:
9101:
8945:
8734:
8524:
8368:
8161:
8058:
8002:
7911:is a CW complex and
7830:simplicial complexes
7675:
7652:
7488:
7395:
7366:
7343:
7298:
7294:-adic cohomology of
7278:
7247:
7119:
6959:
6930:
6910:
6890:
6866:
6834:
6810:
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.
10849:Cohomology theories
10736:Tychonoff's theorem
10731:Poincaré conjecture
10485:General (point-set)
10030:, Proposition 3.38.
9646:elliptic cohomology
9644:Various flavors of
9467:, via a theorem of
9374:oriented cobordism
9306:
9259:
7865:George W. Whitehead
7834:cellular cohomology
7029:
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
10721:De Rham cohomology
10642:Polyhedral complex
10632:Simplicial complex
10418:10.1007/BF02566923
10279:Algebraic Topology
10245:Algebraic Geometry
9846:Quantum cohomology
9836:Motivic cohomology
9771:Deligne cohomology
9741:Bounded cohomology
9731:Algebraic K-theory
9689:
9621:
9582:
9546:
9507:
9453:
9409:
9364:
9319:
9292:
9272:
9245:
9181:
9141:
9021:
8890:
8604:
8536:
8444:
8311:
8100:
8044:
7863:≠ 0. Around 1960,
7838:de Rham cohomology
7782:
7658:
7648:If the subvariety
7635:
7468:
7466:
7378:
7352:
7322:
7284:
7264:
7230:
7099:
7012:
7007:
6942:
6916:
6906:one can construct
6896:
6872:
6849:
6816:
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
10836:
10835:
10625:fundamental group
10241:Hartshorne, Robin
10204:Eilenberg, Samuel
10188:978-3-540-58660-9
9931:, pp. 62–63.
9821:Khovanov homology
9801:Galois cohomology
9766:Cyclic cohomology
9420:complex cobordism
9237:cohomotopy groups
9132:
8988:
8848:
8819:
8783:
8527:
8417:
8275:
8240:
8204:
7875:is a sequence of
7871:By definition, a
7661:{\displaystyle Z}
7224:
7131:
7000:
6899:{\displaystyle p}
6875:{\displaystyle p}
6819:{\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
10856:
10826:
10825:
10799:
10798:
10789:
10779:
10769:
10768:
10757:
10756:
10551:
10464:
10457:
10450:
10441:
10436:
10391:
10361:
10338:
10322:
10304:
10269:
10236:
10208:Steenrod, Norman
10199:
10166:
10143:
10121:
10115:
10109:
10108:
10097:
10091:
10085:
10079:
10073:
10067:
10066:, Section III.2.
10061:
10055:
10049:
10043:
10037:
10031:
10025:
10019:
10013:
10007:
10001:
9995:
9989:
9983:
9977:
9968:
9962:
9956:
9950:
9944:
9943:, Theorem II.29.
9938:
9932:
9926:
9920:
9914:
9908:
9902:
9896:
9886:
9880:
9874:
9831:Local cohomology
9806:Group cohomology
9781:Étale cohomology
9698:
9696:
9695:
9690:
9679:
9678:
9630:
9628:
9627:
9622:
9611:
9610:
9591:
9589:
9588:
9583:
9572:
9571:
9555:
9553:
9552:
9547:
9536:
9535:
9516:
9514:
9513:
9508:
9497:
9496:
9462:
9460:
9459:
9454:
9440:
9439:
9418:
9416:
9415:
9410:
9396:
9395:
9373:
9371:
9370:
9365:
9354:
9353:
9328:
9326:
9325:
9320:
9305:
9300:
9281:
9279:
9278:
9273:
9258:
9253:
9190:
9188:
9187:
9182:
9177:
9176:
9164:
9163:
9151:
9150:
9140:
9113:
9112:
9030:
9028:
9027:
9022:
8999:
8998:
8989:
8987:
8986:
8974:
8957:
8956:
8899:
8897:
8896:
8891:
8865:
8864:
8849:
8841:
8830:
8829:
8820:
8818:
8817:
8805:
8794:
8793:
8784:
8782:
8781:
8769:
8752:
8751:
8660:
8613:
8611:
8610:
8605:
8588:
8587:
8572:
8571:
8559:
8558:
8546:
8545:
8535:
8453:
8451:
8450:
8445:
8428:
8427:
8418:
8416:
8415:
8403:
8380:
8379:
8320:
8318:
8317:
8312:
8292:
8291:
8276:
8268:
8251:
8250:
8241:
8239:
8238:
8226:
8215:
8214:
8205:
8203:
8202:
8190:
8179:
8178:
8156:
8137:
8109:
8107:
8106:
8101:
8054:is homotopic to
8053:
8051:
8050:
8045:
7958:
7818:sheaf cohomology
7791:
7789:
7788:
7783:
7772:
7771:
7750:
7749:
7728:
7727:
7703:
7702:
7687:
7686:
7667:
7665:
7664:
7659:
7644:
7642:
7641:
7636:
7619:
7618:
7591:
7590:
7566:
7565:
7550:
7549:
7528:
7527:
7506:
7505:
7477:
7475:
7474:
7469:
7467:
7442:
7424:
7423:
7387:
7385:
7384:
7379:
7361:
7359:
7358:
7353:
7331:
7329:
7328:
7323:
7318:
7317:
7312:
7293:
7291:
7290:
7285:
7273:
7271:
7270:
7265:
7260:
7239:
7237:
7236:
7231:
7229:
7225:
7223:
7219:
7218:
7217:
7199:
7198:
7184:
7183:
7179:
7178:
7177:
7159:
7158:
7144:
7138:
7132:
7129:
7108:
7106:
7105:
7100:
7098:
7097:
7092:
7086:
7085:
7084:
7083:
7078:
7061:
7060:
7048:
7043:
7028:
7023:
7008:
6992:
6991:
6986:
6971:
6970:
6951:
6949:
6948:
6943:
6925:
6923:
6922:
6917:
6905:
6903:
6902:
6897:
6881:
6879:
6878:
6873:
6858:
6856:
6855:
6850:
6848:
6847:
6842:
6828:Hodge structures
6825:
6823:
6822:
6817:
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:
18:Cohomology class
10864:
10863:
10859:
10858:
10857:
10855:
10854:
10853:
10839:
10838:
10837:
10832:
10763:
10745:
10741:Urysohn's lemma
10702:
10666:
10552:
10543:
10515:low-dimensional
10473:
10468:
10394:
10381:
10371:Springer-Verlag
10364:
10351:
10336:
10326:
10307:
10294:
10272:
10259:
10249:Springer-Verlag
10239:
10226:
10202:
10189:
10179:Springer-Verlag
10169:
10156:
10134:Dieudonné, Jean
10132:
10129:
10124:
10116:
10112:
10099:
10098:
10094:
10086:
10082:
10074:
10070:
10064:Hartshorne 1977
10062:
10058:
10054:, Section IV.3.
10050:
10046:
10038:
10034:
10026:
10022:
10014:
10010:
10002:
9998:
9990:
9986:
9982:, Theorem 3.19.
9978:
9971:
9967:, Theorem 3.15.
9963:
9959:
9955:, Example 3.16.
9951:
9947:
9939:
9935:
9927:
9923:
9919:, Theorem 3.11.
9915:
9911:
9903:
9899:
9891:, Theorem 3.5;
9887:
9883:
9875:
9871:
9867:
9855:
9850:
9791:Flat cohomology
9751:Čech cohomology
9746:BRST cohomology
9723:
9714:
9670:
9665:
9664:
9639:Morava K-theory
9602:
9594:
9593:
9563:
9558:
9557:
9527:
9519:
9518:
9488:
9480:
9479:
9431:
9423:
9422:
9387:
9376:
9375:
9345:
9337:
9336:
9287:
9286:
9240:
9239:
9168:
9155:
9142:
9104:
9099:
9098:
9084:
9075:
9066:
9057:
8990:
8978:
8948:
8943:
8942:
8850:
8821:
8809:
8785:
8773:
8743:
8732:
8731:
8635:
8579:
8563:
8550:
8537:
8522:
8521:
8507:
8498:
8489:
8480:
8419:
8407:
8371:
8366:
8365:
8277:
8242:
8230:
8206:
8194:
8170:
8159:
8158:
8139:
8125:
8056:
8055:
8000:
7999:
7986:
7972:
7952:
7934:
7925:
7919:
7886:
7845:dimension axiom
7810:Čech cohomology
7806:
7798:
7763:
7741:
7719:
7694:
7678:
7673:
7672:
7650:
7649:
7604:
7582:
7557:
7541:
7519:
7497:
7486:
7485:
7465:
7464:
7459:
7454:
7448:
7447:
7441:
7435:
7434:
7415:
7410:
7405:
7393:
7392:
7364:
7363:
7341:
7340:
7307:
7296:
7295:
7276:
7275:
7245:
7244:
7209:
7190:
7189:
7185:
7169:
7150:
7149:
7145:
7139:
7133:
7117:
7116:
7087:
7073:
7068:
7052:
6981:
6962:
6957:
6956:
6928:
6927:
6908:
6907:
6888:
6887:
6864:
6863:
6837:
6832:
6831:
6808:
6807:
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
6314:
6292:
6270:
6265:
6264:
6218:J. W. Alexander
6203:
6152:
6117:
6112:
6111:
6101:
6053:
6032:
5927:
5911:
5883:
5878:
5877:
5816:
5788:
5760:
5755:
5754:
5694:
5666:
5638:
5627:
5626:
5612:
5606:
5551:
5540:
5539:
5489:
5484:
5483:
5482:. For example,
5445:
5440:
5439:
5400:
5386:
5385:
5349:
5344:
5343:
5316:covering spaces
5279:
5274:
5273:
5241:
5230:
5229:
5208:
5203:
5202:
5160:
5155:
5154:
5133:
5128:
5127:
5091:
5090:
5048:
5047:
5004:
4953:
4952:
4905:
4904:
4895:on every space
4841:
4836:
4835:
4781:
4780:
4769:
4763:
4734:submanifold of
4667:
4661:
4635:with values in
4601:perfect pairing
4555:
4521:
4493:
4488:
4487:
4449:
4443:
4388:
4363:
4338:
4318:
4317:
4198:
4170:
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
3601:
3591:
3566:
3532:
3527:
3526:
3504:Künneth formula
3415:
3405:
3397:
3396:
3363:
3358:
3357:
3320:
3310:
3302:
3301:
3276:polynomial ring
3242:
3237:
3236:
3195:
3138:
3133:
3132:
3096:
3085:
3084:
3048:
3043:
3042:
3005:
3000:
2999:
2969:
2941:
2936:
2935:
2881:
2876:
2875:
2833:
2828:
2827:
2778:
2777:
2756:
2755:
2608:
2564:
2563:
2558:that intersect
2516:
2448:closed manifold
2396:
2368:
2355:
2350:
2349:
2312:
2311:
2270:
2236:
2207:
2184:
2183:
2171:cohomology ring
2129:
2091:
2086:
2085:
2017:
1989:
1961:
1956:
1955:
1906:Noetherian ring
1845:
1840:
1839:
1788:
1783:
1782:
1739:
1734:
1733:
1673:
1655:
1627:
1579:
1545:
1544:
1483:
1461:
1439:
1411:
1400:
1399:
1346:
1341:
1340:
1292:
1264:
1242:
1220:
1198:
1187:
1186:
1107:
1085:
1072:
1067:
1066:
1032:
1010:
997:
992:
991:
956:
955:
807:
806:
776:
765:
741:For an integer
682:
641:
600:
599:
597:cochain complex
522:
517:
516:
492:
487:
486:
453:
416:
415:
409:
392:
379:
352:
307:
290:
271:
251:
226:
215:
214:
200:homotopy groups
154:
127:
109:
56:cochain complex
36:homology theory
28:
23:
22:
15:
12:
11:
5:
10862:
10860:
10852:
10851:
10841:
10840:
10834:
10833:
10831:
10830:
10820:
10819:
10818:
10813:
10808:
10793:
10783:
10773:
10761:
10750:
10747:
10746:
10744:
10743:
10738:
10733:
10728:
10723:
10718:
10712:
10710:
10704:
10703:
10701:
10700:
10695:
10690:
10688:Winding number
10685:
10680:
10674:
10672:
10668:
10667:
10665:
10664:
10659:
10654:
10649:
10644:
10639:
10634:
10629:
10628:
10627:
10622:
10620:homotopy group
10612:
10611:
10610:
10605:
10600:
10595:
10590:
10580:
10575:
10570:
10560:
10558:
10554:
10553:
10546:
10544:
10542:
10541:
10536:
10531:
10530:
10529:
10519:
10518:
10517:
10507:
10502:
10497:
10492:
10487:
10481:
10479:
10475:
10474:
10469:
10467:
10466:
10459:
10452:
10444:
10438:
10437:
10392:
10379:
10362:
10349:
10324:
10305:
10292:
10274:Hatcher, Allen
10270:
10257:
10237:
10224:
10200:
10187:
10171:Dold, Albrecht
10167:
10154:
10128:
10125:
10123:
10122:
10110:
10092:
10080:
10068:
10056:
10052:Dieudonné 1989
10044:
10042:, p. 177.
10032:
10020:
10018:, p. 186.
10008:
10006:, Example 3.7.
9996:
9994:, p. 222.
9984:
9969:
9957:
9945:
9933:
9921:
9909:
9897:
9889:Hatcher (2001)
9881:
9879:, p. 108.
9868:
9866:
9863:
9862:
9861:
9854:
9851:
9849:
9848:
9843:
9838:
9833:
9828:
9823:
9818:
9813:
9808:
9803:
9798:
9796:Floer homology
9793:
9788:
9783:
9778:
9773:
9768:
9763:
9758:
9753:
9748:
9743:
9738:
9733:
9727:
9722:
9719:
9712:
9688:
9685:
9682:
9677:
9673:
9661:multiplicative
9659:is said to be
9650:
9649:
9642:
9632:
9620:
9617:
9614:
9609:
9605:
9601:
9581:
9578:
9575:
9570:
9566:
9545:
9542:
9539:
9534:
9530:
9526:
9506:
9503:
9500:
9495:
9491:
9487:
9472:
9469:Daniel Quillen
9452:
9449:
9446:
9443:
9438:
9434:
9430:
9408:
9405:
9402:
9399:
9394:
9390:
9386:
9383:
9363:
9360:
9357:
9352:
9348:
9344:
9329:
9318:
9315:
9312:
9309:
9304:
9299:
9295:
9271:
9268:
9265:
9262:
9257:
9252:
9248:
9197:
9196:
9180:
9175:
9171:
9167:
9162:
9158:
9154:
9149:
9145:
9139:
9135:
9131:
9128:
9125:
9122:
9119:
9116:
9111:
9107:
9080:
9071:
9062:
9053:
9036:
9020:
9017:
9014:
9011:
9008:
9005:
9002:
8997:
8993:
8985:
8981:
8977:
8972:
8969:
8966:
8963:
8960:
8955:
8951:
8900:
8889:
8886:
8883:
8880:
8877:
8874:
8871:
8868:
8863:
8860:
8857:
8853:
8847:
8844:
8839:
8836:
8833:
8828:
8824:
8816:
8812:
8808:
8803:
8800:
8797:
8792:
8788:
8780:
8776:
8772:
8767:
8764:
8761:
8758:
8755:
8750:
8746:
8742:
8739:
8689:
8630:(for integers
8620:
8619:
8603:
8600:
8597:
8594:
8591:
8586:
8582:
8578:
8575:
8570:
8566:
8562:
8557:
8553:
8549:
8544:
8540:
8534:
8530:
8503:
8494:
8485:
8476:
8459:
8443:
8440:
8437:
8434:
8431:
8426:
8422:
8414:
8410:
8406:
8401:
8398:
8395:
8392:
8389:
8386:
8383:
8378:
8374:
8321:
8310:
8307:
8304:
8301:
8298:
8295:
8290:
8287:
8284:
8280:
8274:
8271:
8266:
8263:
8260:
8257:
8254:
8249:
8245:
8237:
8233:
8229:
8224:
8221:
8218:
8213:
8209:
8201:
8197:
8193:
8188:
8185:
8182:
8177:
8173:
8169:
8166:
8111:
8099:
8096:
8093:
8090:
8087:
8084:
8081:
8078:
8075:
8072:
8069:
8066:
8063:
8043:
8040:
8037:
8034:
8031:
8028:
8025:
8022:
8019:
8016:
8013:
8010:
8007:
7981:
7967:
7947:
7930:
7921:
7887:(for integers
7882:
7859:) = 0 for all
7797:
7794:
7793:
7792:
7781:
7778:
7775:
7770:
7766:
7762:
7759:
7756:
7753:
7748:
7744:
7740:
7737:
7734:
7731:
7726:
7722:
7718:
7715:
7712:
7709:
7706:
7701:
7697:
7693:
7690:
7685:
7681:
7657:
7646:
7645:
7634:
7631:
7628:
7625:
7622:
7617:
7614:
7611:
7607:
7603:
7600:
7597:
7594:
7589:
7585:
7581:
7578:
7575:
7572:
7569:
7564:
7560:
7556:
7553:
7548:
7544:
7540:
7537:
7534:
7531:
7526:
7522:
7518:
7515:
7512:
7509:
7504:
7500:
7496:
7493:
7479:
7478:
7463:
7460:
7458:
7455:
7453:
7450:
7449:
7446:
7443:
7440:
7437:
7436:
7433:
7430:
7427:
7422:
7418:
7414:
7411:
7409:
7406:
7404:
7401:
7400:
7377:
7374:
7371:
7351:
7348:
7321:
7316:
7311:
7306:
7303:
7283:
7263:
7259:
7255:
7252:
7241:
7240:
7228:
7222:
7216:
7212:
7208:
7205:
7202:
7197:
7193:
7188:
7182:
7176:
7172:
7168:
7165:
7162:
7157:
7153:
7148:
7143:
7136:
7127:
7124:
7110:
7109:
7096:
7091:
7082:
7077:
7071:
7067:
7064:
7059:
7055:
7051:
7047:
7042:
7038:
7035:
7032:
7027:
7022:
7019:
7015:
7011:
7006:
7003:
6998:
6995:
6990:
6985:
6980:
6977:
6974:
6969:
6965:
6941:
6938:
6935:
6915:
6895:
6884:étale topology
6871:
6846:
6841:
6815:
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:
24:
14:
13:
10:
9:
6:
4:
3:
2:
10861:
10850:
10847:
10846:
10844:
10829:
10821:
10817:
10814:
10812:
10809:
10807:
10804:
10803:
10802:
10794:
10792:
10788:
10784:
10782:
10778:
10774:
10772:
10767:
10762:
10760:
10752:
10751:
10748:
10742:
10739:
10737:
10734:
10732:
10729:
10727:
10724:
10722:
10719:
10717:
10714:
10713:
10711:
10709:
10705:
10699:
10698:Orientability
10696:
10694:
10691:
10689:
10686:
10684:
10681:
10679:
10676:
10675:
10673:
10669:
10663:
10660:
10658:
10655:
10653:
10650:
10648:
10645:
10643:
10640:
10638:
10635:
10633:
10630:
10626:
10623:
10621:
10618:
10617:
10616:
10613:
10609:
10606:
10604:
10601:
10599:
10596:
10594:
10591:
10589:
10586:
10585:
10584:
10581:
10579:
10576:
10574:
10571:
10569:
10565:
10562:
10561:
10559:
10555:
10550:
10540:
10537:
10535:
10534:Set-theoretic
10532:
10528:
10525:
10524:
10523:
10520:
10516:
10513:
10512:
10511:
10508:
10506:
10503:
10501:
10498:
10496:
10495:Combinatorial
10493:
10491:
10488:
10486:
10483:
10482:
10480:
10476:
10472:
10465:
10460:
10458:
10453:
10451:
10446:
10445:
10442:
10435:
10431:
10427:
10423:
10419:
10415:
10411:
10407:
10406:
10401:
10397:
10393:
10390:
10386:
10382:
10380:3-540-42750-3
10376:
10372:
10368:
10363:
10360:
10356:
10352:
10350:0-226-51182-0
10346:
10342:
10335:
10334:
10329:
10328:May, J. Peter
10325:
10320:
10316:
10315:
10310:
10306:
10303:
10299:
10295:
10293:0-521-79540-0
10289:
10285:
10281:
10280:
10275:
10271:
10268:
10264:
10260:
10258:0-387-90244-9
10254:
10250:
10246:
10242:
10238:
10235:
10231:
10227:
10225:9780691627236
10221:
10217:
10213:
10209:
10205:
10201:
10198:
10194:
10190:
10184:
10180:
10176:
10172:
10168:
10165:
10161:
10157:
10155:0-8176-3388-X
10151:
10147:
10142:
10141:
10135:
10131:
10130:
10126:
10119:
10114:
10111:
10106:
10102:
10096:
10093:
10089:
10084:
10081:
10078:, p. 95.
10077:
10072:
10069:
10065:
10060:
10057:
10053:
10048:
10045:
10041:
10036:
10033:
10029:
10024:
10021:
10017:
10012:
10009:
10005:
10000:
9997:
9993:
9988:
9985:
9981:
9976:
9974:
9970:
9966:
9961:
9958:
9954:
9949:
9946:
9942:
9937:
9934:
9930:
9925:
9922:
9918:
9913:
9910:
9906:
9901:
9898:
9894:
9890:
9885:
9882:
9878:
9873:
9870:
9864:
9860:
9857:
9856:
9852:
9847:
9844:
9842:
9839:
9837:
9834:
9832:
9829:
9827:
9824:
9822:
9819:
9817:
9814:
9812:
9809:
9807:
9804:
9802:
9799:
9797:
9794:
9792:
9789:
9787:
9784:
9782:
9779:
9777:
9774:
9772:
9769:
9767:
9764:
9762:
9759:
9757:
9754:
9752:
9749:
9747:
9744:
9742:
9739:
9737:
9734:
9732:
9729:
9728:
9726:
9720:
9718:
9716:
9715:ring spectrum
9711:
9707:, such as an
9706:
9705:ring spectrum
9702:
9683:
9675:
9671:
9662:
9658:
9653:
9647:
9643:
9640:
9636:
9633:
9615:
9607:
9603:
9599:
9576:
9568:
9564:
9540:
9532:
9528:
9524:
9501:
9493:
9489:
9485:
9477:
9473:
9470:
9466:
9465:formal groups
9450:
9444:
9436:
9432:
9428:
9421:
9406:
9400:
9392:
9388:
9384:
9381:
9358:
9350:
9346:
9342:
9334:
9330:
9316:
9310:
9302:
9297:
9293:
9285:
9269:
9263:
9255:
9250:
9246:
9238:
9234:
9233:
9232:
9229:
9226:
9221:
9219:
9215:
9209:
9206:
9202:
9194:
9173:
9169:
9165:
9160:
9156:
9147:
9143:
9137:
9133:
9123:
9120:
9117:
9109:
9105:
9096:
9095:product group
9092:
9088:
9083:
9079:
9074:
9070:
9065:
9061:
9056:
9052:
9048:
9044:
9040:
9037:
9034:
9015:
9012:
9009:
9006:
9003:
8995:
8991:
8983:
8979:
8967:
8964:
8961:
8953:
8949:
8940:
8936:
8932:
8928:
8924:
8920:
8916:
8912:
8908:
8904:
8901:
8887:
8884:
8875:
8872:
8869:
8861:
8858:
8855:
8851:
8845:
8834:
8826:
8822:
8814:
8810:
8798:
8790:
8786:
8778:
8774:
8762:
8759:
8756:
8748:
8744:
8737:
8729:
8725:
8721:
8717:
8713:
8709:
8705:
8701:
8697:
8694:: Each pair (
8693:
8690:
8687:
8684:
8683:
8682:
8680:
8676:
8672:
8668:
8664:
8658:
8654:
8650:
8646:
8642:
8638:
8633:
8629:
8625:
8617:
8598:
8595:
8592:
8584:
8580:
8568:
8564:
8560:
8555:
8551:
8542:
8538:
8532:
8528:
8519:
8515:
8511:
8506:
8502:
8497:
8493:
8488:
8484:
8479:
8475:
8471:
8467:
8463:
8460:
8457:
8438:
8435:
8432:
8424:
8420:
8412:
8408:
8396:
8393:
8390:
8387:
8384:
8376:
8372:
8363:
8359:
8355:
8351:
8347:
8343:
8339:
8335:
8331:
8327:
8326:
8322:
8308:
8305:
8296:
8288:
8285:
8282:
8278:
8261:
8258:
8255:
8247:
8243:
8235:
8231:
8219:
8211:
8207:
8199:
8195:
8183:
8175:
8171:
8164:
8154:
8150:
8146:
8142:
8136:
8132:
8128:
8123:
8119:
8116:: Each pair (
8115:
8112:
8094:
8091:
8088:
8076:
8073:
8070:
8064:
8061:
8038:
8035:
8032:
8020:
8017:
8014:
8008:
8005:
7997:
7994:
7993:
7992:
7990:
7984:
7980:
7976:
7970:
7966:
7962:
7956:
7950:
7946:
7942:
7938:
7933:
7929:
7924:
7918:
7914:
7910:
7906:
7902:
7898:
7894:
7890:
7885:
7881:
7878:
7874:
7869:
7866:
7862:
7858:
7854:
7850:
7846:
7841:
7839:
7835:
7831:
7827:
7823:
7819:
7815:
7811:
7805:
7800:
7795:
7776:
7768:
7764:
7760:
7754:
7746:
7742:
7738:
7732:
7724:
7720:
7716:
7707:
7699:
7695:
7691:
7683:
7679:
7671:
7670:
7669:
7655:
7632:
7623:
7615:
7612:
7609:
7605:
7595:
7587:
7583:
7570:
7562:
7558:
7554:
7546:
7542:
7538:
7532:
7524:
7520:
7510:
7502:
7498:
7491:
7484:
7483:
7482:
7461:
7451:
7428:
7420:
7416:
7412:
7402:
7391:
7390:
7389:
7375:
7372:
7369:
7349:
7346:
7337:
7335:
7314:
7301:
7281:
7250:
7226:
7220:
7214:
7210:
7206:
7203:
7200:
7195:
7191:
7186:
7180:
7174:
7170:
7166:
7163:
7160:
7155:
7151:
7146:
7134:
7125:
7122:
7115:
7114:
7113:
7094:
7080:
7069:
7057:
7053:
7045:
7036:
7033:
7025:
7020:
7017:
7013:
7009:
7004:
7001:
6996:
6988:
6978:
6975:
6967:
6963:
6955:
6954:
6953:
6939:
6936:
6933:
6913:
6893:
6885:
6869:
6860:
6844:
6829:
6813:
6800:
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:
6498:
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:
6200:
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:
5603:
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:
4760:
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:
4658:
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:
4440:
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:
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4287:
4283:
4279:
4275:
4271:
4267:
4263:
4259:
4255:
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4242:
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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:
4017:
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:
3192:
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:
151:
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:
10828:Publications
10693:Chern number
10683:Betti number
10566: /
10557:Key concepts
10526:
10505:Differential
10409:
10403:
10366:
10332:
10312:
10309:"Cohomology"
10278:
10244:
10211:
10174:
10139:
10118:Switzer 1975
10113:
10105:MathOverflow
10104:
10095:
10088:Switzer 1975
10083:
10071:
10059:
10047:
10035:
10028:Hatcher 2001
10023:
10016:Hatcher 2001
10011:
10004:Hatcher 2001
9999:
9992:Hatcher 2001
9987:
9980:Hatcher 2001
9965:Hatcher 2001
9960:
9953:Hatcher 2001
9948:
9936:
9924:
9917:Hatcher 2001
9912:
9900:
9884:
9877:Hatcher 2001
9872:
9724:
9709:
9700:
9660:
9656:
9654:
9651:
9230:
9222:
9218:triangulated
9214:phantom maps
9210:
9198:
9192:
9090:
9086:
9081:
9077:
9072:
9068:
9063:
9059:
9054:
9050:
9046:
9042:
9038:
9032:
8938:
8934:
8930:
8926:
8922:
8918:
8914:
8910:
8906:
8902:
8727:
8723:
8719:
8715:
8711:
8707:
8703:
8699:
8695:
8691:
8685:
8678:
8674:
8670:
8666:
8662:
8656:
8652:
8648:
8644:
8640:
8636:
8631:
8627:
8623:
8621:
8615:
8513:
8509:
8504:
8500:
8495:
8491:
8486:
8482:
8477:
8473:
8469:
8465:
8461:
8455:
8361:
8357:
8353:
8349:
8345:
8341:
8337:
8333:
8329:
8323:
8152:
8148:
8144:
8140:
8134:
8130:
8126:
8121:
8117:
8113:
7995:
7988:
7982:
7978:
7974:
7968:
7964:
7960:
7954:
7948:
7944:
7940:
7936:
7931:
7927:
7922:
7912:
7908:
7904:
7900:
7888:
7883:
7879:
7872:
7870:
7860:
7856:
7852:
7848:
7844:
7842:
7807:
7799:
7647:
7480:
7338:
7242:
7111:
6861:
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:
10791:Wikiversity
10708:Key results
9893:Dold (1972)
8661:called the
7959:called the
7891:) from the
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
10637:CW complex
10578:Continuity
10568:Closed set
10527:cohomology
10396:Thom, René
10146:Birkhäuser
10127:References
9786:Ext groups
9191:for every
9039:Additivity
9031:for every
8614:for every
8518:direct sum
8462:Additivity
8454:for every
7802:See also:
7362: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
10816:geometric
10811:algebraic
10662:Cobordism
10598:Hausdorff
10593:connected
10510:Geometric
10500:Continuum
10490:Algebraic
10434:120243638
10412:: 17–86,
10319:EMS Press
9941:Thom 1954
9929:Thom 1954
9905:Dold 1972
9865:Citations
9676:∗
9608:∗
9569:∗
9533:∗
9494:∗
9437:∗
9393:∗
9351:∗
9333:cobordism
9298:∗
9294:π
9256:∗
9247:π
9174:α
9161:α
9138:α
9134:∏
9130:→
9013:∩
8984:∗
8976:→
8885:⋯
8882:→
8843:→
8815:∗
8807:→
8779:∗
8771:→
8741:→
8738:⋯
8692:Exactness
8665:(writing
8577:→
8569:α
8556:α
8533:α
8529:⨁
8413:∗
8405:→
8394:∩
8306:⋯
8303:→
8286:−
8273:∂
8270:→
8236:∗
8228:→
8200:∗
8192:→
8168:→
8165:⋯
8114:Exactness
8083:→
8027:→
7761:⊕
7739:≅
7717:⊕
7633:⋯
7630:→
7602:→
7580:→
7539:⊕
7517:→
7495:→
7492:⋯
7457:⟶
7445:↓
7439:↓
7408:⟶
7373:⊂
7347:≥
7282:ℓ
7204:…
7164:…
7095:ℓ
7081:ℓ
7070:⊗
7054:ℓ
7010:
7005:←
6989:ℓ
6937:≠
6934:ℓ
6914:ℓ
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
10843:Category
10781:Wikibook
10759:Category
10647:Manifold
10615:Homotopy
10573:Interior
10564:Open set
10522:Homology
10471:Topology
10398:(1954),
10330:(1999),
10276:(2001),
10243:(1977),
10210:(1952),
10173:(1972),
10136:(1989),
10076:May 1999
10040:May 1999
9853:See also
9476:K-theory
9201:spectrum
8903:Excision
8686:Homotopy
8325:Excision
7996:Homotopy
7893:category
7877:functors
7274: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
10806:general
10608:uniform
10588:compact
10539:Digital
10426:0061823
10389:0385836
10359:1702278
10321:, 2001
10302:1867354
10267:0463157
10234:0050886
10197:0415602
10164:0995842
10120:, 7.68.
9235:Stable
8722:,∅) → (
8147:,∅) → (
7903:,
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
10801:Topics
10603:metric
10478:Fields
10432:
10424:
10387:
10377:
10357:
10347:
10300:
10290:
10265:
10255:
10232:
10222:
10195:
10185:
10162:
10152:
9041:: If (
8673:) for
8464:: If (
7963:(here
7907:) (so
7895:of CW-
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
10583:Space
10430:S2CID
10337:(PDF)
9205:Adams
9085:) → (
8933:) → (
8905:: If
8508:) → (
8356:) → (
8328:: If
7998:: If
7897:pairs
7847:: 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(
10375:ISBN
10345:ISBN
10288:ISBN
10253:ISBN
10220:ISBN
10183:ISBN
10150:ISBN
8913:and
8714:and
8647:) →
8336:and
8138:and
7943:) →
7828:for
7130:Proj
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
10414:doi
9663:if
8921:: (
8730:):
8718:: (
8344:: (
8143:: (
7816:or
7002:lim
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
10845::
10428:,
10422:MR
10420:,
10410:28
10408:,
10402:,
10385:MR
10383:,
10373:,
10369:,
10355:MR
10353:,
10343:,
10339:,
10317:,
10311:,
10298:MR
10296:,
10286:,
10282:,
10263:MR
10261:,
10251:,
10230:MR
10228:,
10218:,
10214:,
10206:;
10193:MR
10191:,
10181:,
10177:,
10160:MR
10158:,
10148:,
10144:,
10103:.
9972:^
9637:,
9220:.
9199:A
9097::
8710:→
8706::
8639::
8520::
8157::
8133:→
8129::
7985:−1
7971:−1
7951:−1
7939:,
7926::
7832:,
7812:,
6997::=
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:,
10463:e
10456:t
10449:v
10416::
10323:.
10107:.
9713:∞
9710:E
9701:X
9687:)
9684:X
9681:(
9672:E
9657:E
9648:.
9619:)
9616:X
9613:(
9604:u
9600:k
9580:)
9577:X
9574:(
9565:K
9544:)
9541:X
9538:(
9529:o
9525:k
9505:)
9502:X
9499:(
9490:O
9486:K
9471:.
9451:,
9448:)
9445:X
9442:(
9433:U
9429:M
9407:,
9404:)
9401:X
9398:(
9389:O
9385:S
9382:M
9362:)
9359:X
9356:(
9347:O
9343:M
9317:.
9314:)
9311:X
9308:(
9303:S
9270:.
9267:)
9264:X
9261:(
9251:S
9195:.
9193:i
9179:)
9170:A
9166:,
9157:X
9153:(
9148:i
9144:h
9127:)
9124:A
9121:,
9118:X
9115:(
9110:i
9106:h
9091:A
9089:,
9087:X
9082:α
9078:A
9076:,
9073:α
9069:X
9064:α
9060:A
9058:,
9055:α
9051:X
9047:A
9045:,
9043:X
9035:.
9033:i
9019:)
9016:B
9010:A
9007:,
9004:A
9001:(
8996:i
8992:h
8980:f
8971:)
8968:B
8965:,
8962:X
8959:(
8954:i
8950:h
8939:B
8937:,
8935:X
8931:B
8929:∩
8927:A
8925:,
8923:A
8919:f
8915:B
8911:A
8907:X
8888:.
8879:)
8876:A
8873:,
8870:X
8867:(
8862:1
8859:+
8856:i
8852:h
8846:d
8838:)
8835:A
8832:(
8827:i
8823:h
8811:f
8802:)
8799:X
8796:(
8791:i
8787:h
8775:g
8766:)
8763:A
8760:,
8757:X
8754:(
8749:i
8745:h
8728:A
8726:,
8724:X
8720:X
8716:g
8712:X
8708:A
8704:f
8700:A
8698:,
8696:X
8679:A
8677:(
8675:h
8671:A
8669:(
8667:h
8659:)
8657:A
8655:,
8653:X
8651:(
8649:h
8645:A
8643:(
8641:h
8637:d
8632:i
8628:h
8618:.
8616:i
8602:)
8599:A
8596:,
8593:X
8590:(
8585:i
8581:h
8574:)
8565:A
8561:,
8552:X
8548:(
8543:i
8539:h
8514:A
8512:,
8510:X
8505:α
8501:A
8499:,
8496:α
8492:X
8487:α
8483:A
8481:,
8478:α
8474:X
8470:A
8468:,
8466:X
8458:.
8456:i
8442:)
8439:B
8436:,
8433:X
8430:(
8425:i
8421:h
8409:f
8400:)
8397:B
8391:A
8388:,
8385:A
8382:(
8377:i
8373:h
8362:B
8360:,
8358:X
8354:B
8352:∩
8350:A
8348:,
8346:A
8342:f
8338:B
8334:A
8330:X
8309:.
8300:)
8297:A
8294:(
8289:1
8283:i
8279:h
8265:)
8262:A
8259:,
8256:X
8253:(
8248:i
8244:h
8232:g
8223:)
8220:X
8217:(
8212:i
8208:h
8196:f
8187:)
8184:A
8181:(
8176:i
8172:h
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8151:,
8149:X
8145:X
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8122:A
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8098:)
8095:B
8092:,
8089:Y
8086:(
8080:)
8077:A
8074:,
8071:X
8068:(
8065::
8062:g
8042:)
8039:B
8036:,
8033:Y
8030:(
8024:)
8021:A
8018:,
8015:X
8012:(
8009::
8006:f
7989:A
7987:(
7983:i
7979:h
7975:A
7973:(
7969:i
7965:h
7957:)
7955:A
7953:(
7949:i
7945:h
7941:A
7937:X
7935:(
7932:i
7928:h
7923:i
7920:∂
7913:A
7909:X
7905:A
7901:X
7899:(
7889:i
7884:i
7880:h
7861:i
7857:P
7855:(
7853:H
7849:P
7780:)
7777:E
7774:(
7769:n
7765:H
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7752:(
7747:n
7743:H
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7733:Z
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7503:n
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7315:q
7310:F
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7221:)
7215:k
7211:f
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7201:,
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7181:]
7175:n
7171:x
7167:,
7161:,
7156:0
7152:x
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7142:Z
7135:(
7126:=
7123:X
7090:Q
7076:Z
7066:)
7063:)
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7050:(
7046:/
7041:Z
7037:;
7034:X
7031:(
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6979:;
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6726:,
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6719:R
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6699:⊗
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6687:(
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