Cohomology

Inmathematics,specifically inhomology theoryandalgebraic topology,cohomologyis a general term for a sequence ofabelian groups,usually one associated with atopological space,often defined from acochain complex.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 arefunctionson the group ofchainsin homology theory.

From its start intopology,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 throughoutgeometryandalgebra.The terminology tends to hide the fact that cohomology, acontravarianttheory, is more natural than homology in many applications. At a basic level, this has to do with functions andpullbacksin geometric situations: given spaces$X$and$Y$,and some function$F$on$Y$,for anymapping$f:X\to Y$,composition with$f$gives rise to a function$F\circ f$on$X$.The most important cohomology theories have a product, thecup product,which gives them aringstructure. Because of this feature, cohomology is usually a stronger invariant than homology.

Singular cohomology

Singular cohomologyis a powerful invariant in topology, associating agraded-commutative ringwith any topological space. Everycontinuous map$f:X\to Y$determines ahomomorphismfrom the cohomology ring of$Y$to that of$X$;this puts strong restrictions on the possible maps from$X$to$Y$.Unlike more subtle invariants such ashomotopy groups,the cohomology ring tends to be computable in practice for spaces of interest.

For a topological space$X$,the definition of singular cohomology starts with thesingular chain complex: $$\cdots \to C_{i+1}{\stackrel {\partial _{i+1}}{\to }}C_{i}{\stackrel {\partial _{i}}{\to }}\ C_{i-1}\to \cdots$$ By definition, thesingular homologyof$X$is the homology of this chain complex (the kernel of one homomorphism modulo the image of the previous one). In more detail,$C_{i}$is thefree abelian groupon the set of continuous maps from the standard$i$-simplex to$X$(called "singular$i$-simplices in$X$"), and$\partial _{i}$is the$i$-th boundary homomorphism. The groups$C_{i}$are zero for$i$negative.

Now fix an abelian group$A$,and replace each group$C_{i}$by itsdual group$C_{i}^{*}=\mathrm {Hom} (C_{i},A),$and$\partial _{i}$by itsdual homomorphism $$d_{i-1}:C_{i-1}^{*}\to C_{i}^{*}.$$

This has the effect of "reversing all the arrows" of the original complex, leaving acochain complex $$\cdots \leftarrow C_{i+1}^{*}{\stackrel {d_{i}}{\leftarrow }}\ C_{i}^{*}{\stackrel {d_{i-1}}{\leftarrow }}C_{i-1}^{*}\leftarrow \cdots$$

For an integer$i$,the$i$^thcohomology groupof$X$with coefficients in$A$is defined to be$\operatorname {ker} (d_{i})/\operatorname {im} (d_{i-1})$and denoted by$H^{i}(X,A)$.The group$H^{i}(X,A)$is zero for$i$negative. The elements of$C_{i}^{*}$are calledsingular$i$-cochainswith coefficients in$A$.(Equivalently, an$i$-cochain on$X$can be identified with a function from the set of singular$i$-simplices in$X$to$A$.) Elements of$\ker(d)$and${\textrm {im}}(d)$are calledcocyclesandcoboundaries,respectively, while elements of$\operatorname {ker} (d_{i})/\operatorname {im} (d_{i-1})=H^{i}(X,A)$are calledcohomology classes(because they areequivalence classesof cocycles).

In what follows, the coefficient group$A$is sometimes not written. It is common to take$A$to be acommutative ring$R$;then the cohomology groups are$R$-modules.A standard choice is the ring$\mathbb {Z}$ofintegers.

Some of the formal properties of cohomology are only minor variants of the properties of homology:

• A continuous map$f:X\to Y$determines apushforwardhomomorphism$f_{*}:H_{i}(X)\to H_{i}(Y)$on homology and apullbackhomomorphism$f^{*}:H^{i}(Y)\to H^{i}(X)$on cohomology. This makes cohomology into acontravariant functorfrom topological spaces to abelian groups (or$R$-modules).
• Twohomotopicmaps from$X$to$Y$induce the same homomorphism on cohomology (just as on homology).
• TheMayer–Vietoris sequenceis 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$X$is the union ofopen subsets$U$and$V$,then there is along exact sequence:$$\cdots \to H^{i}(X)\to H^{i}(U)\oplus H^{i}(V)\to H^{i}(U\cap V)\to H^{i+1}(X)\to \cdots$$
• There arerelative cohomologygroups$H^{i}(X,Y;A)$for anysubspace$Y$of a space$X$.They are related to the usual cohomology groups by a long exact sequence:$$\cdots \to H^{i}(X,Y)\to H^{i}(X)\to H^{i}(Y)\to H^{i+1}(X,Y)\to \cdots$$
• Theuniversal coefficient theoremdescribes cohomology in terms of homology, usingExt groups.Namely, there is ashort exact sequence$$0\to \operatorname {Ext} _{\mathbb {Z} }^{1}(\operatorname {H} _{i-1}(X,\mathbb {Z} ),A)\to H^{i}(X,A)\to \operatorname {Hom} _{\mathbb {Z} }(H_{i}(X,\mathbb {Z} ),A)\to 0.$$A related statement is that for afield$F$,$H^{i}(X,F)$is precisely thedual spaceof thevector space$H_{i}(X,F)$.
• If$X$is a topologicalmanifoldor aCW complex,then the cohomology groups$H^{i}(X,A)$are zero for$i$greater than thedimensionof$X$.If$X$is acompactmanifold (possibly with boundary), or a CW complex with finitely many cells in each dimension, and$R$is a commutativeNoetherian ring,then the$R$-module$H^{i}(X,R)$isfinitely generatedfor each$i$.

On the other hand, cohomology has a crucial structure that homology does not: for any topological space$X$and commutative ring$R$,there is abilinear map,called thecup product: $$H^{i}(X,R)\times H^{j}(X,R)\to H^{i+j}(X,R),$$ defined by an explicit formula on singular cochains. The product of cohomology classes$u$and$v$is written as$u\cup v$or simply as$uv$.This product makes thedirect sum $$H^{*}(X,R)=\bigoplus _{i}H^{i}(X,R)$$ into agraded ring,called thecohomology ringof$X$.It isgraded-commutativein the sense that: $$uv=(-1)^{ij}vu,\qquad u\in H^{i}(X,R),v\in H^{j}(X,R).$$

For any continuous map$f\colon X\to Y,$the pullback$f^{*}:H^{*}(Y,R)\to H^{*}(X,R)$is a homomorphism of graded$R$-algebras.It follows that if two spaces arehomotopy equivalent,then their cohomology rings are isomorphic.

Here are some of the geometric interpretations of the cup product. In what follows,manifoldsare understood to be without boundary, unless stated otherwise. Aclosed manifoldmeans a compact manifold (without boundary), whereas a closedsubmanifoldNof a manifoldMmeans a submanifold that is aclosed subsetofM,not necessarily compact (althoughNis automatically compact ifMis).

• LetXbe a closedorientedmanifold of dimensionn.ThenPoincaré dualitygives an isomorphismHⁱX≅H_n−iX.As a result, a closed oriented submanifoldSofcodimensioniinXdetermines a cohomology class inHⁱX,called [S]. In these terms, the cup product describes the intersection of submanifolds. Namely, ifSandTare submanifolds of codimensioniandjthat intersecttransversely,then$$[S][T]=[S\cap T]\in H^{i+j}(X),$$where the intersectionS∩Tis a submanifold of codimensioni+j,with an orientation determined by the orientations ofS,T,andX.In the case ofsmooth manifolds,ifSandTdo not intersect transversely, this formula can still be used to compute the cup product [S][T], by perturbingSorTto make the intersection transverse.
  More generally, without assuming thatXhas an orientation, a closed submanifold ofXwith an orientation on itsnormal bundledetermines a cohomology class onX.IfXis a noncompact manifold, then a closed submanifold (not necessarily compact) determines a cohomology class onX.In both cases, the cup product can again be described in terms of intersections of submanifolds.
  Note thatThomconstructed 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.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.
• For a smooth manifoldX,de Rham's theoremsays that the singular cohomology ofXwithrealcoefficients is isomorphic to the de Rham cohomology ofX,defined usingdifferential forms.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 tochain homotopy.In fact, it is impossible to modify the definition of singular cochains with coefficients in the integers$\mathbb {Z}$or in$\mathbb {Z} /p$for a prime numberpto make the product graded-commutative on the nose. The failure of graded-commutativity at the cochain level leads to theSteenrod operationson modpcohomology.

Very informally, for any topological spaceX,elements of$H^{i}(X)$can be thought of as represented by codimension-isubspaces ofXthat can move freely onX.For example, one way to define an element of$H^{i}(X)$is to give a continuous mapffromXto a manifoldMand a closed codimension-isubmanifoldNofMwith an orientation on the normal bundle. Informally, one thinks of the resulting class$f^{*}([N])\in H^{i}(X)$as lying on the subspace$f^{-1}(N)$ofX;this is justified in that the class$f^{*}([N])$restricts to zero in the cohomology of the open subset$X-f^{-1}(N).$The cohomology class$f^{*}([N])$can move freely onXin the sense thatNcould be replaced by any continuous deformation ofNinsideM.

Examples

In what follows, cohomology is taken with coefficients in the integersZ,unless stated otherwise.

• The cohomology ring of a point is the ringZin degree 0. By homotopy invariance, this is also the cohomology ring of anycontractiblespace, such as Euclidean spaceRⁿ.

• 

  The first cohomology group of the 2-dimensional torus has a basis given by the classes of the two circles shown.

  For a positive integern,the cohomology ring of thesphere$S^{n}$isZ[x]/(x²) (thequotient ringof apolynomial ringby the givenideal), withxin degreen.In terms of Poincaré duality as above,xis the class of a point on the sphere.
• The cohomology ring of thetorus$(S^{1})^{n}$is theexterior algebraoverZonngenerators in degree 1.For example, letPdenote a point in the circle$S^{1}$,andQthe point (P,P) in the 2-dimensional torus$(S^{1})^{2}$.Then the cohomology of (S¹)²has a basis as afreeZ-moduleof the form: the element 1 in degree 0,x:= [P×S¹] andy:= [S¹×P] in degree 1, andxy= [Q] in degree 2. (Implicitly, orientations of the torus and of the two circles have been fixed here.) Note thatyx= −xy= −[Q], by graded-commutativity.
• More generally, letRbe a commutative ring, and letXandYbe any topological spaces such thatH^*(X,R) is a finitely generated freeR-module in each degree. (No assumption is needed onY.) Then theKünneth formulagives that the cohomology ring of theproduct spaceX×Yis atensor productofR-algebras:$$H^{*}(X\times Y,R)\cong H^{*}(X,R)\otimes _{R}H^{*}(Y,R).$$
• The cohomology ring ofreal projective spaceRPⁿwithZ/2 coefficients isZ/2[x]/(xⁿ⁺¹), withxin degree 1.Herexis the class of ahyperplaneRPⁿ⁻¹inRPⁿ;this makes sense even thoughRP^jis not orientable forjeven and positive, because Poincaré duality withZ/2 coefficients works for arbitrary manifolds.
  With integer coefficients, the answer is a bit more complicated. TheZ-cohomology ofRP^2ahas an elementyof degree 2 such that the whole cohomology is the direct sum of a copy ofZspanned by the element 1 in degree 0 together with copies ofZ/2 spanned by the elementsyⁱfori=1,...,a.TheZ-cohomology ofRP^2a+1is the same together with an extra copy ofZin degree 2a+1.
• The cohomology ring ofcomplex projective spaceCPⁿisZ[x]/(xⁿ⁺¹), withxin degree 2.Herexis the class of a hyperplaneCPⁿ⁻¹inCPⁿ.More generally,x^jis the class of a linear subspaceCP^n−jinCPⁿ.
• The cohomology ring of the closed oriented surfaceXofgenusg≥ 0 has a basis as a freeZ-module of the form: the element 1 in degree 0,A₁,...,A_gandB₁,...,B_gin degree 1, and the classPof a point in degree 2. The product is given by:A_iA_j=B_iB_j= 0 for alliandj,A_iB_j= 0 ifi≠j,andA_iB_i=Pfor alli.By graded-commutativity, it follows thatB_iA_i= −P.
• On any topological space, graded-commutativity of the cohomology ring implies that 2x²= 0 for all odd-degree cohomology classesx.It follows that for a ringRcontaining 1/2, all odd-degree elements ofH^*(X,R) have square zero. On the other hand, odd-degree elements need not have square zero ifRisZ/2 orZ,as one sees in the example ofRP²(withZ/2 coefficients) orRP⁴×RP²(withZcoefficients).

The diagonal

The cup product on cohomology can be viewed as coming from thediagonal map$\Delta:X\to X\times X$,$x\mapsto (x,x)$.Namely, for any spaces$X$and$Y$with cohomology classes$u\in H^{i}(X,R)$and$v\in H^{j}(Y,R)$,there is anexternal product(orcross product) cohomology class$u\times v\in H^{i+j}(X\times Y,R)$.The cup product of classes$u\in H^{i}(X,R)$and$v\in H^{j}(X,R)$can be defined as the pullback of the external product by the diagonal: $$uv=\Delta ^{*}(u\times v)\in H^{i+j}(X,R).$$

Alternatively, the external product can be defined in terms of the cup product. For spaces$X$and$Y$,write$f:X\times Y\to X$and$g:X\times Y\to Y$for the two projections. Then the external product of classes$u\in H^{i}(X,R)$and$v\in H^{j}(Y,R)$is: $$u\times v=(f^{*}(u))(g^{*}(v))\in H^{i+j}(X\times Y,R).$$

Poincaré duality

Main article:Poincaré duality

Another interpretation of Poincaré duality is that the cohomology ring of a closed oriented manifold is self-dual in a strong sense. Namely, let$X$be a closedconnectedoriented manifold of dimension$n$,and let$F$be a field. Then$H^{n}(X,F)$is isomorphic to$F$,and the product

$H^{i}(X,F)\times H^{n-i}(X,F)\to H^{n}(X,F)\cong F$

is aperfect pairingfor each integer$i$.In particular, the vector spaces$H^{i}(X,F)$and$H^{n-i}(X,F)$have the same (finite) dimension. Likewise, the product on integral cohomology modulotorsionwith values in$H^{n}(X,\mathbb {Z} )\cong \mathbb {Z}$is a perfect pairing over$\mathbb {Z}$.

Characteristic classes

Main article:Characteristic class

An oriented realvector bundleEof rankrover a topological spaceXdetermines a cohomology class onX,theEuler classχ(E) ∈H^r(X,Z). Informally, the Euler class is the class of the zero set of a generalsectionofE.That interpretation can be made more explicit whenEis a smooth vector bundle over a smooth manifoldX,since then a general smooth section ofXvanishes on a codimension-rsubmanifold ofX.

There are several other types ofcharacteristic classesfor vector bundles that take values in cohomology, includingChern classes,Stiefel–Whitney classes,andPontryagin classes.

Eilenberg–MacLane spaces

Main article:Eilenberg–MacLane space

For each abelian groupAand natural numberj,there is a space$K(A,j)$whosej-th homotopy group is isomorphic toAand whose other homotopy groups are zero. Such a space is called anEilenberg–MacLane space.This space has the remarkable property that it is aclassifying spacefor cohomology: there is a natural elementuof$H^{j}(K(A,j),A)$,and every cohomology class of degreejon every spaceXis the pullback ofuby some continuous map$X\to K(A,j)$.More precisely, pulling back the classugives a bijection

$[X,K(A,j)]{\stackrel {\cong }{\to }}H^{j}(X,A)$

for every spaceXwith the homotopy type of a CW complex.Here$[X,Y]$denotes the set of homotopy classes of continuous maps fromXtoY.

For example, the space$K(\mathbb {Z},1)$(defined up to homotopy equivalence) can be taken to be the circle$S^{1}$.So the description above says that every element of$H^{1}(X,\mathbb {Z} )$is pulled back from the classuof a point on$S^{1}$by some map$X\to S^{1}$.

There is a related description of the first cohomology with coefficients in any abelian groupA,say for a CW complexX.Namely,$H^{1}(X,A)$is in one-to-one correspondence with the set of isomorphism classes of Galoiscovering spacesofXwith groupA,also calledprincipalA-bundlesoverX.ForXconnected, it follows that$H^{1}(X,A)$is isomorphic to$\operatorname {Hom} (\pi _{1}(X),A)$,where$\pi _{1}(X)$is thefundamental groupofX.For example,$H^{1}(X,\mathbb {Z} /2)$classifies the double covering spaces ofX,with the element$0\in H^{1}(X,\mathbb {Z} /2)$corresponding to the trivial double covering, the disjoint union of two copies ofX.

Cap product

Main article:Cap product

For any topological spaceX,thecap productis a bilinear map

$\cap:H^{i}(X,R)\times H_{j}(X,R)\to H_{j-i}(X,R)$

for any integersiandjand any commutative ringR.The resulting map

$H^{*}(X,R)\times H_{*}(X,R)\to H_{*}(X,R)$

makes the singular homology ofXinto a module over the singular cohomology ring ofX.

Fori=j,the cap product gives the natural homomorphism

$H^{i}(X,R)\to \operatorname {Hom} _{R}(H_{i}(X,R),R),$

which is an isomorphism forRa field.

For example, letXbe an oriented manifold, not necessarily compact. Then a closed oriented codimension-isubmanifoldYofX(not necessarily compact) determines an element ofHⁱ(X,R), and a compact orientedj-dimensional submanifoldZofXdetermines an element ofH_j(X,R). The cap product [Y] ∩ [Z] ∈H_j−i(X,R) can be computed by perturbingYandZto make them intersect transversely and then taking the class of their intersection, which is a compact oriented submanifold of dimensionj−i.

A closed oriented manifoldXof dimensionnhas afundamental class[X] inH_n(X,R). The Poincaré duality isomorphism $$H^{i}(X,R){\overset {\cong }{\to }}H_{n-i}(X,R)$$ is defined by cap product with the fundamental class ofX.

Brief history of singular cohomology

Although cohomology is fundamental to modern algebraic topology, its importance was not seen for some 40 years after the development of homology. The concept ofdual cell structure,whichHenri Poincaréused in his proof of his Poincaré duality theorem, contained the beginning of the idea of cohomology, but this was not seen until later.

There were various precursors to cohomology.In the mid-1920s,J. W. AlexanderandSolomon Lefschetzfoundedintersection theoryof cycles on manifolds. On a closed orientedn-dimensional manifoldMani-cycle and aj-cycle with nonempty intersection will, if in thegeneral position,have as their intersection a (i+j−n)-cycle. This leads to a multiplication of homology classes

$H_{i}(M)\times H_{j}(M)\to H_{i+j-n}(M),$

which (in retrospect) can be identified with thecup producton the cohomology ofM.

Alexander had by 1930 defined a first notion of a cochain, by thinking of ani-cochain on a spaceXas a function on small neighborhoods of the diagonal inXⁱ⁺¹.

In 1931,Georges de Rhamrelated homology and differential forms, provingde Rham's theorem.This result can be stated more simply in terms of cohomology.

In 1934,Lev Pontryaginproved thePontryagin dualitytheorem; a result ontopological groups.This (in rather special cases) provided an interpretation of Poincaré duality andAlexander dualityin terms of groupcharacters.

At a 1935 conference inMoscow,Andrey Kolmogorovand Alexander both introduced cohomology and tried to construct a cohomology product structure.

In 1936,Norman SteenrodconstructedČech cohomologyby dualizing Čech homology.

From 1936 to 1938,Hassler WhitneyandEduard Čechdeveloped thecup product(making cohomology into a graded ring) andcap product,and realized that Poincaré duality can be stated in terms of the cap product. Their theory was still limited to finite cell complexes.

In 1944,Samuel Eilenbergovercame the technical limitations, and gave the modern definition of singular homology and cohomology.

In 1945, Eilenberg and Steenrod stated theaxiomsdefining a homology or cohomology theory, discussed below. In their 1952 book,Foundations of Algebraic Topology,they proved that the existing homology and cohomology theories did indeed satisfy their axioms.

In 1946,Jean Leraydefined sheaf cohomology.

In 1948Edwin Spanier,building on work of Alexander and Kolmogorov, developedAlexander–Spanier cohomology.

Sheaf cohomology

Main article:Sheaf cohomology

Sheaf cohomologyis a rich generalization of singular cohomology, allowing more general "coefficients" than simply an abelian group. For everysheafof abelian groupsEon a topological spaceX,one has cohomology groupsHⁱ(X,E) for integersi.In particular, in the case of theconstant sheafonXassociated with an abelian groupA,the resulting groupsHⁱ(X,A) coincide with singular cohomology forXa manifold or CW complex (though not for arbitrary spacesX). Starting in the 1950s, sheaf cohomology has become a central part ofalgebraic geometryandcomplex analysis,partly because of the importance of the sheaf ofregular functionsor the sheaf ofholomorphic functions.

Grothendieckelegantly defined and characterized sheaf cohomology in the language ofhomological algebra.The essential point is to fix the spaceXand think of sheaf cohomology as a functor from theabelian categoryof sheaves onXto abelian groups. Start with the functor taking a sheafEonXto its abelian group of global sections overX,E(X). This functor isleft exact,but not necessarily right exact. Grothendieck defined sheaf cohomology groups to be the rightderived functorsof the left exact functorE↦E(X).

That definition suggests various generalizations. For example, one can define the cohomology of a topological spaceXwith coefficients in any complex of sheaves, earlier calledhypercohomology(but usually now just "cohomology" ). From that point of view, sheaf cohomology becomes a sequence of functors from thederived categoryof sheaves onXto abelian groups.

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 ringR,theTor groupsTor_i^R(M,N) form a "homology theory" in each variable, the left derived functors of the tensor productM⊗_RNofR-modules. Likewise, theExt groupsExtⁱ_R(M,N) can be viewed as a "cohomology theory" in each variable, the right derived functors of the Hom functor Hom_R(M,N).

Sheaf cohomology can be identified with a type of Ext group. Namely, for a sheafEon a topological spaceX,Hⁱ(X,E) is isomorphic to Extⁱ(Z_X,E), whereZ_Xdenotes the constant sheaf associated with the integersZ,and Ext is taken in the abelian category of sheaves onX.

Cohomology of varieties

There are numerous machines built for computing the cohomology ofalgebraic varieties.The simplest case being the determination of cohomology forsmoothprojective varietiesover a field ofcharacteristic$0$.Tools fromHodge theory,calledHodge structures,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 smoothhypersurfacein$\mathbb {P} ^{n}$can be determined from the degree of the polynomial alone.

When considering varieties over afinite field,or a field of characteristic$p$,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 aGrothendieck topologyand used sheaf cohomology over theétale topologyto define the cohomology theory for varieties over a finite field. Using the étale topology for a variety over a field of characteristic$p$one can construct$\ell$-adic cohomologyfor$\ell \neq p$.This is defined as theprojective limit

$H^{k}(X;\mathbb {Q} _{\ell }):=\varprojlim _{n\in \mathbb {N} }H_{et}^{k}(X;\mathbb {Z} /(\ell ^{n}))\otimes _{\mathbb {Z} _{\ell }}\mathbb {Q} _{\ell }.$

If we have a scheme of finite type

$X=\operatorname {Proj} \left({\frac {\mathbb {Z} \left[x_{0},\ldots,x_{n}\right]}{\left(f_{1},\ldots,f_{k}\right)}}\right)$

then there is an equality of dimensions for the Betti cohomology of$X(\mathbb {C} )$and the$\ell$-adic cohomology of$X(\mathbb {F} _{q})$whenever the variety is smooth over both fields. In addition to these cohomology theories there are other cohomology theories calledWeil cohomology theorieswhich behave similarly to singular cohomology. There is a conjectured theory of motives which underlie all of the Weil cohomology theories.

Another useful computational tool is the blowup sequence. Given a codimension$\geq 2$subscheme$Z\subset X$there is aCartesian square

${\begin{matrix}E&\longrightarrow &Bl_{Z}(X)\\\downarrow &&\downarrow \\Z&\longrightarrow &X\end{matrix}}$

From this there is an associated long exact sequence

$\cdots \to H^{n}(X)\to H^{n}(Z)\oplus H^{n}(Bl_{Z}(X))\to H^{n}(E)\to H^{n+1}(X)\to \cdots$

If the subvariety$Z$is smooth, then the connecting morphisms are all trivial, hence

$H^{n}(Bl_{Z}(X))\oplus H^{n}(Z)\cong H^{n}(X)\oplus H^{n}(E)$

Axioms and generalized cohomology theories

See also:List of cohomology theories

There are various ways to define cohomology for topological spaces (such as singular cohomology,Čech cohomology,Alexander–Spanier cohomologyorsheaf cohomology). (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 theEilenberg–Steenrod axioms,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 assimplicial cohomologyforsimplicial complexes,cellular cohomologyfor CW complexes, andde Rham cohomologyfor smooth manifolds.

One of the Eilenberg–Steenrod axioms for a cohomology theory is thedimension axiom:ifPis a single point, thenHⁱ(P) = 0 for alli≠ 0. Around 1960,George W. Whiteheadobserved 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 space, not directly accessible from singular cohomology. (In this context, singular cohomology is often called "ordinary cohomology".)

By definition, ageneralized homology theoryis a sequence offunctorsh_i(for integersi) from thecategoryof CW-pairs(X,A) (soXis a CW complex andAis a subcomplex) to the category of abelian groups, together with anatural transformation∂_i:h_i(X,A) →h_i−1(A)called theboundary homomorphism(hereh_i−1(A) is a shorthand forh_i−1(A,∅)). The axioms are:

1. Homotopy:If$f:(X,A)\to (Y,B)$is homotopic to$g:(X,A)\to (Y,B)$,then the induced homomorphisms on homology are the same.
2. Exactness:Each pair (X,A) induces a long exact sequence in homology, via the inclusionsf:A→Xandg:(X,∅) → (X,A):$$\cdots \to h_{i}(A){\overset {f_{*}}{\to }}h_{i}(X){\overset {g_{*}}{\to }}h_{i}(X,A){\overset {\partial }{\to }}h_{i-1}(A)\to \cdots.$$
3. Excision:IfXis the union of subcomplexesAandB,then the inclusionf:(A,A∩B) → (X,B) induces an isomorphism$$h_{i}(A,A\cap B){\overset {f_{*}}{\to }}h_{i}(X,B)$$for everyi.
4. Additivity:If (X,A) is the disjoint union of a set of pairs (X_α,A_α), then the inclusions (X_α,A_α) → (X,A) induce an isomorphism from thedirect sum:$$\bigoplus _{\alpha }h_{i}(X_{\alpha },A_{\alpha })\to h_{i}(X,A)$$for everyi.

The axioms for a generalized cohomology theory are obtained by reversing the arrows, roughly speaking. In more detail, ageneralized cohomology theoryis a sequence of contravariant functorshⁱ(for integersi) from the category of CW-pairs to the category of abelian groups, together with a natural transformationd:hⁱ(A) →hⁱ⁺¹(X,A)called theboundary homomorphism(writinghⁱ(A) forhⁱ(A,∅)). The axioms are:

1. Homotopy:Homotopic maps induce the same homomorphism on cohomology.
2. Exactness:Each pair (X,A) induces a long exact sequence in cohomology, via the inclusionsf:A→Xandg:(X,∅) → (X,A):$$\cdots \to h^{i}(X,A){\overset {g_{*}}{\to }}h^{i}(X){\overset {f_{*}}{\to }}h^{i}(A){\overset {d}{\to }}h^{i+1}(X,A)\to \cdots.$$
3. Excision:IfXis the union of subcomplexesAandB,then the inclusionf:(A,A∩B) → (X,B) induces an isomorphism$$h^{i}(X,B){\overset {f_{*}}{\to }}h^{i}(A,A\cap B)$$for everyi.
4. Additivity:If (X,A) is the disjoint union of a set of pairs (X_α,A_α), then the inclusions (X_α,A_α) → (X,A) induce an isomorphism to theproduct group:$$h^{i}(X,A)\to \prod _{\alpha }h^{i}(X_{\alpha },A_{\alpha })$$for everyi.

Aspectrumdetermines both a generalized homology theory and a generalized cohomology theory. A fundamental result by Brown, Whitehead, andAdamssays 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.

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 (calledphantom maps) 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 beingtriangulated.

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 everyweak homotopy equivalenceinduces 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 corresponding theory on CW complexes.

Some examples of generalized cohomology theories are:

• Stablecohomotopy groups$\pi _{S}^{*}(X).$The corresponding homology theory is used more often:stable homotopy groups$\pi _{*}^{S}(X).$
• Various different flavors ofcobordismgroups, based on studying a space by considering all maps from it to manifolds: unoriented cobordism$MO^{*}(X)$oriented cobordism$MSO^{*}(X),$complex cobordism$MU^{*}(X),$and so on. Complex cobordism has turned out to be especially powerful in homotopy theory. It is closely related toformal groups,via a theorem ofDaniel Quillen.
• Various different flavors of topologicalK-theory,based on studying a space by considering all vector bundles over it:$KO^{*}(X)$(real periodic K-theory),$ko^{*}(X)$(real connective K-theory),$K^{*}(X)$(complex periodic K-theory),$ku^{*}(X)$(complex connective K-theory), and so on.
• Brown–Peterson cohomology,Morava K-theory,Morava E-theory, and other theories built from complex cobordism.
• Various flavors ofelliptic cohomology.

Many of these theories carry richer information than ordinary cohomology, but are harder to compute.

A cohomology theoryEis said to bemultiplicativeif$E^{*}(X)$has the structure of a graded ring for each spaceX.In the language of spectra, there are several more precise notions of aring spectrum,such as anE_∞ring spectrum,where the product is commutative and associative in a strong sense.

Other cohomology theories

Cohomology theories in a broader sense (invariants of other algebraic or geometric structures, rather than of topological spaces) include:

• Algebraic K-theory
• André–Quillen cohomology
• Bounded cohomology
• BRST cohomology
• Čech cohomology
• Coherent sheaf cohomology
• Crystalline cohomology
• Cyclic cohomology
• Deligne cohomology
• Equivariant cohomology
• Étale cohomology
• Ext groups
• Flat cohomology
• Floer homology
• Galois cohomology
• Group cohomology
• Hochschild cohomology
• Intersection cohomology
• Khovanov homology
• Lie algebra cohomology
• Local cohomology
• Motivic cohomology
• Non-abelian cohomology
• Quantum cohomology

See also

• complex-oriented cohomology theory

References

• Dieudonné, Jean(1989),History of Algebraic and Differential Topology,Birkhäuser,ISBN0-8176-3388-X,MR0995842
• Dold, Albrecht(1972),Lectures on Algebraic Topology,Springer-Verlag,ISBN978-3-540-58660-9,MR0415602
• Eilenberg, Samuel;Steenrod, Norman(1952),Foundations of Algebraic Topology,Princeton University Press,ISBN9780691627236,MR0050886
• Hartshorne, Robin(1977),Algebraic Geometry,Graduate Texts in Mathematics, vol. 52, New York, Heidelberg:Springer-Verlag,ISBN0-387-90244-9,MR0463157
• Hatcher, Allen(2001),Algebraic Topology,Cambridge University Press,ISBN0-521-79540-0,MR1867354
• "Cohomology",Encyclopedia of Mathematics,EMS Press,2001 [1994].
• May, J. Peter(1999),A Concise Course in Algebraic Topology(PDF),University of Chicago Press,ISBN0-226-51182-0,MR1702278
• Switzer, Robert (1975),Algebraic Topology — Homology and Homotopy,Springer-Verlag,ISBN3-540-42750-3,MR0385836
• Thom, René(1954),"Quelques propriétés globales des variétés différentiables",Commentarii Mathematici Helvetici,28:17–86,doi:10.1007/BF02566923,MR0061823,S2CID120243638