View source for Cohomology

{{Short description|Sequences of abelian groups attached with topological spaces}}
{{Use American English|date=January 2019}}
In [[mathematics]], specifically in [[homology theory]] and [[algebraic topology]], '''cohomology''' is a general term for a sequence of [[abelian group]]s, usually one associated with a [[topological space]], often defined from a [[cochain 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 are [[function (mathematics)|function]]s on the group of [[chain (algebraic topology)|chains]] in homology theory.

From its start in [[topology]], 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 [[geometry]] and [[abstract algebra|algebra]]. The terminology tends to hide the fact that cohomology, a [[Covariance and contravariance of functors|contravariant]] theory, is more natural than homology in many applications. At a basic level, this has to do with functions and [[pullback (differential geometry)|pullback]]s in geometric situations: given spaces <math>X</math> and <math>Y</math>, and some function <math>F</math> on <math>Y</math>, for any [[Map (mathematics)|mapping]] <math>f:X\to Y</math>, composition with <math>f</math> gives rise to a function <math>F\circ f</math> on <math>X</math>. The most important cohomology theories have a product, the [[cup product]], which gives them a [[ring (mathematics)|ring]] structure.  Because of this feature, cohomology is usually a stronger invariant than homology.

==Singular cohomology<!--'Singular cohomology' redirects here-->==
'''Singular cohomology'''<!--boldface per WP:R#PLA--> is a powerful invariant in topology, associating a [[graded-commutative ring]] with any topological space. Every [[continuous map]] <math>f:X\to Y</math> determines a [[ring homomorphism|homomorphism]] from the cohomology ring of <math>Y</math> to that of <math>X</math>; this puts strong restrictions on the possible maps from <math>X</math> to <math>Y</math>. Unlike more subtle invariants such as [[homotopy group]]s, the cohomology ring tends to be computable in practice for spaces of interest.

For a topological space <math>X</math>, the definition of singular cohomology starts with the [[singular chain complex]]:{{sfn|Hatcher|2001|p=108}}
<math display="block">\cdots \to C_{i+1}\stackrel{\partial_{i+1}}{\to} C_i \stackrel{ \partial_i}{\to}\ C_{i-1} \to \cdots </math>
By definition, the [[singular homology]] of <math>X</math> is the homology of this chain complex (the kernel of one homomorphism modulo the image of the previous one). In more detail, <math>C_i</math> is the [[free abelian group]] on the set of continuous maps from the standard <math>i</math>-simplex to <math>X</math> (called "singular <math>i</math>-simplices in <math>X</math>"), and <math>\partial_i</math> is the <math>i</math>-th boundary homomorphism. The groups <math>C_i</math> are zero for <math>i</math> negative.

Now fix an abelian group <math>A</math>, and replace each group <math>C_i</math> by its [[dual space|dual group]] <math>C_i^* = \mathrm{Hom}(C_i,A),</math> and <math>\partial_i</math> by its [[dual space#Transpose of a linear map|dual homomorphism]]
<math display="block">d_{i-1}: C_{i-1}^* \to C_{i}^*.</math>

This has the effect of "reversing all the arrows" of the original complex, leaving a [[cochain complex]]
<math display="block">\cdots \leftarrow C_{i+1}^* \stackrel{d_i}{\leftarrow}\ C_{i}^* \stackrel{d_{i-1}}{\leftarrow} C_{i-1}^* \leftarrow \cdots </math>

For an integer <math>i</math>, the <math>i</math><sup>th</sup> '''cohomology group''' of <math>X</math> with coefficients in <math>A</math> is defined to be <math>\operatorname{ker}(d_i)/\operatorname{im}(d_{i-1})</math> and denoted by <math>H^i(X,A)</math>. The group <math>H^i(X,A)</math> is zero for <math>i</math> negative. The elements of <math>C_i^*</math> are called '''singular <math>i</math>-cochains''' with coefficients in <math>A</math>. (Equivalently, an <math>i</math>-cochain on <math>X</math> can be identified with a function from the set of singular <math>i</math>-simplices in <math>X</math> to <math>A</math>.) Elements of <math>\ker(d)</math> and <math>\textrm{im}(d)</math> are called '''cocycles''' and '''coboundaries''', respectively, while elements of <math>\operatorname{ker}(d_i)/\operatorname{im}(d_{i-1})=H^i(X,A)</math> are called '''cohomology classes''' (because they are [[equivalence class]]es of cocycles).

In what follows, the coefficient group <math>A</math> is sometimes not written. It is common to take <math>A</math> to be a [[commutative ring]] <math>R</math>; then the cohomology groups are <math>R</math>-[[module (mathematics)|module]]s. A standard choice is the ring <math>\Z</math> of [[integer]]s.

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

* A continuous map <math>f: X \to Y</math> determines a [[Pushforward (homology)|'''pushforward''']] homomorphism <math>f_*:H_i(X) \to H_i(Y)</math> on homology and a [[pullback (cohomology)|'''pullback''']] homomorphism <math>f^*: H^i(Y) \to H^i(X)</math> on cohomology. This makes cohomology into a [[contravariant functor]] from topological spaces to abelian groups (or <math>R</math>-modules).
* Two [[homotopic]] maps from <math>X</math> to <math>Y</math> induce the same homomorphism on cohomology (just as on homology).
* The [[Mayer–Vietoris sequence]] 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 <math>X</math> is the union of [[open subset]]s <math>U</math> and <math>V</math>, then there is a [[long exact sequence]]: <math display="block">\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</math>
* There are [[relative homology|relative cohomology]] groups <math>H^i(X,Y;A)</math> for any [[subspace topology|subspace]] <math>Y</math> of a space <math>X</math>. They are related to the usual cohomology groups by a long exact sequence: <math display="block">\cdots \to H^i(X,Y) \to H^i(X) \to H^i(Y) \to H^{i+1}(X,Y) \to \cdots</math>
* The [[universal coefficient theorem]] describes cohomology in terms of homology, using [[Ext group]]s. Namely, there is a [[short exact sequence]] <math display="block"> 0 \to \operatorname{Ext}_{\Z}^1(\operatorname{H}_{i-1}(X, \Z), A) \to H^i(X, A) \to \operatorname{Hom}_{\Z}(H_i(X,\Z), A)\to 0.</math> A related statement is that for a [[field (mathematics)|field]] <math>F</math>, <math>H^i(X,F)</math> is precisely the [[dual space]] of the [[vector space]] <math>H_i(X,F)</math>.
* If <math>X</math> is a topological [[manifold]] or a [[CW complex]], then the cohomology groups <math>H^i(X,A)</math> are zero for <math>i</math> greater than the [[dimension]] of <math>X</math>.{{sfnmp|1a1=Hatcher|1y=2001|2a1=Dold|2y=1972|1loc=Theorem 3.5|2loc=Proposition VIII.3.3 and Corollary VIII.3.4}} If <math>X</math> is a [[compact space|compact]] manifold (possibly with boundary), or a CW complex with finitely many cells in each dimension, and <math>R</math> is a commutative [[Noetherian ring]], then the <math>R</math>-module <math>H^i(X,R)</math> is [[finitely generated module|finitely generated]] for each <math>i</math>.{{sfn|Dold|1972|loc=Propositions IV.8.12 and V.4.11}}

On the other hand, cohomology has a crucial structure that homology does not: for any topological space <math>X</math> and commutative ring <math>R</math>, there is a [[bilinear map]], called the '''[[cup product]]''':
<math display="block">H^i(X,R)\times H^j(X,R) \to H^{i+j}(X,R),</math>
defined by an explicit formula on singular cochains. The product of cohomology classes <math>u</math> and <math>v</math> is written as <math>u\cup v</math> or simply as <math>uv</math>. This product makes the [[direct sum]]
<math display="block">H^*(X,R)=\bigoplus_i H^i(X,R)</math>
into a [[graded ring]], called the '''[[cohomology ring]]''' of <math>X</math>. It is [[graded-commutative]] in the sense that:{{sfn|Hatcher|2001|loc=Theorem 3.11}}
<math display="block">uv=(-1)^{ij}vu, \qquad u \in H^i(X,R), v \in H^j(X,R).</math>

For any continuous map <math>f\colon X\to Y,</math> the pullback <math>f^*: H^*(Y,R) \to H^*(X, R)</math> is a homomorphism of graded <math>R</math>-[[associative algebra|algebra]]s. It follows that if two spaces are [[homotopy equivalent]], then their cohomology rings are isomorphic.

Here are some of the geometric interpretations of the cup product. In what follows, [[manifold]]s are understood to be without boundary, unless stated otherwise. A [[closed manifold]] means a compact manifold (without boundary), whereas a closed ''submanifold'' ''N'' of a manifold ''M'' means a submanifold that is a [[closed subset]] of ''M'', not necessarily compact (although ''N'' is automatically compact if ''M'' is).

* Let ''X'' be a closed [[orientability|oriented]] manifold of dimension ''n''. Then [[Poincaré duality]] gives an isomorphism ''H''<sup>''i''</sup>''X'' ≅ ''H''<sub>''n''−''i''</sub>''X''. As a result, a closed oriented submanifold ''S'' of [[codimension]] ''i'' in ''X'' determines a cohomology class in ''H''<sup>''i''</sup>''X'', called [''S'']. In these terms, the cup product describes the intersection of submanifolds. Namely, if ''S'' and ''T'' are submanifolds of codimension ''i'' and ''j'' that intersect [[transversality (mathematics)|transversely]], then <math display="block">[S][T]=[S\cap T]\in H^{i+j}(X),</math> where the intersection ''S'' ∩ ''T'' is a submanifold of codimension ''i'' + ''j'', with an orientation determined by the orientations of ''S'', ''T'', and ''X''. In the case of [[smooth manifold]]s, if ''S'' and ''T'' do not intersect transversely, this formula can still be used to compute the cup product [''S''][''T''], by perturbing ''S'' or ''T'' to make the intersection transverse.{{pb}} More generally, without assuming that ''X'' has an orientation, a closed submanifold of ''X'' with an orientation on its [[normal bundle]] determines a cohomology class on ''X''. If ''X'' is a noncompact manifold, then a closed submanifold (not necessarily compact) determines a cohomology class on ''X''. In both cases, the cup product can again be described in terms of intersections of submanifolds.{{pb}} Note that [[René Thom|Thom]] constructed an integral cohomology class of degree 7 on a smooth 14-manifold that is not the class of any smooth submanifold.{{sfn|Thom|1954|pp=62–63}} 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.{{sfn|Thom|1954|loc=Theorem II.29}} 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 manifold ''X'', [[de Rham's theorem]] says that the singular cohomology of ''X'' with [[real number|real]] coefficients is isomorphic to the de Rham cohomology of ''X'', defined using [[differential form]]s. 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 [[chain homotopy]]. In fact, it is impossible to modify the definition of singular cochains with coefficients in the integers <math>\Z</math> or in <math>\Z/p</math> for a prime number ''p'' to make the product graded-commutative on the nose. The failure of graded-commutativity at the cochain level leads to the [[Steenrod operation]]s on mod ''p'' cohomology.

Very informally, for any topological space ''X'', elements of <math>H^i(X)</math> can be thought of as represented by codimension-''i'' subspaces of ''X'' that can move freely on ''X''. For example, one way to define an element of <math>H^i(X)</math> is to give a continuous map ''f'' from ''X'' to a manifold ''M'' and a closed codimension-''i'' submanifold ''N'' of ''M'' with an orientation on the normal bundle. Informally, one thinks of the resulting class <math>f^*([N]) \in H^i(X)</math> as lying on the subspace <math>f^{-1}(N)</math> of ''X''; this is justified in that the class <math>f^*([N])</math> restricts to zero in the cohomology of the open subset <math>X-f^{-1}(N).</math> The cohomology class <math>f^*([N])</math> can move freely on ''X'' in the sense that ''N'' could be replaced by any continuous deformation of ''N'' inside ''M''.

==Examples==
In what follows, cohomology is taken with coefficients in the integers '''Z''', unless stated otherwise. 
*The cohomology ring of a point is the ring '''Z''' in degree 0. By homotopy invariance, this is also the cohomology ring of any [[contractible]] space, such as Euclidean space '''R'''<sup>''n''</sup>.
*[[File:torus cycles.svg|thumb|right|The first cohomology group of the 2-dimensional torus has a basis given by the classes of the two circles shown.]]For a positive integer ''n'', the cohomology ring of the [[n-sphere|sphere]] <math>S^n</math> is '''Z'''[''x'']/(''x''<sup>2</sup>) (the [[quotient ring]] of a [[polynomial ring]] by the given [[ideal (ring theory)|ideal]]), with ''x'' in degree ''n''. In terms of Poincaré duality as above, ''x'' is the class of a point on the sphere.
*The cohomology ring of the [[torus]] <math>(S^1)^n</math> is the [[exterior algebra]] over '''Z''' on ''n'' generators in degree 1.{{sfn|Hatcher|2001|loc=Example 3.16}} For example, let ''P'' denote a point in the circle <math>S^1</math>, and ''Q'' the point (''P'',''P'') in the 2-dimensional torus <math>(S^1)^2</math>. Then the cohomology of (''S''<sup>1</sup>)<sup>2</sup> has a basis as a [[free module|free '''Z'''-module]] of the form: the element 1 in degree 0, ''x'' := [''P'' × ''S''<sup>1</sup>] and ''y'' := [''S''<sup>1</sup> × ''P''] in degree 1, and ''xy'' = [''Q''] in degree 2. (Implicitly, orientations of the torus and of the two circles have been fixed here.) Note that ''yx'' = −''xy'' = −[''Q''], by graded-commutativity.
*More generally, let ''R'' be a commutative ring, and let ''X'' and ''Y'' be any topological spaces such that ''H''<sup>*</sup>(''X'',''R'') is a finitely generated free ''R''-module in each degree. (No assumption is needed on ''Y''.) Then the [[Künneth formula]] gives that the cohomology ring of the [[product space]] ''X'' × ''Y'' is a [[tensor product of algebras|tensor product]] of ''R''-algebras:{{sfn|Hatcher|2001|loc=Theorem 3.15}} <math display="block">H^*(X\times Y,R)\cong H^*(X,R)\otimes_R H^*(Y,R).</math>
* The cohomology ring of [[real projective space]] '''RP'''<sup>''n''</sup> with '''Z'''/2 coefficients is '''Z'''/2[''x'']/(''x''<sup>''n''+1</sup>), with ''x'' in degree 1.{{sfn|Hatcher|2001|loc=Theorem 3.19}} Here ''x'' is the class of a [[hyperplane]] '''RP'''<sup>''n''−1</sup> in '''RP'''<sup>''n''</sup>; this makes sense even though '''RP'''<sup>''j''</sup> is not orientable for ''j'' even and positive, because Poincaré duality with '''Z'''/2 coefficients works for arbitrary manifolds.{{pb}} With integer coefficients, the answer is a bit more complicated. The '''Z'''-cohomology of '''RP'''<sup>2''a''</sup> has an element ''y'' of degree 2 such that the whole cohomology is the direct sum of a copy of '''Z''' spanned by the element 1 in degree 0 together with copies of '''Z'''/2 spanned by the elements ''y''<sup>''i''</sup> for ''i''=1,...,''a''. The '''Z'''-cohomology of '''RP'''<sup>2''a''+1</sup> is the same together with an extra copy of '''Z''' in degree 2''a''+1.{{sfn|Hatcher|2001|p=222}}
*The cohomology ring of [[complex projective space]] '''CP'''<sup>''n''</sup> is '''Z'''[''x'']/(''x''<sup>''n''+1</sup>), with ''x'' in degree 2.{{sfn|Hatcher|2001|loc=Theorem 3.19}} Here ''x'' is the class of a hyperplane '''CP'''<sup>''n''−1</sup> in '''CP'''<sup>''n''</sup>. More generally, ''x''<sup>''j''</sup> is the class of a linear subspace '''CP'''<sup>''n''−''j''</sup> in '''CP'''<sup>''n''</sup>.
*The cohomology ring of the closed oriented surface ''X'' of [[genus (mathematics)|genus]] ''g'' ≥ 0 has a basis as a free '''Z'''-module of the form: the element 1 in degree 0, ''A''<sub>1</sub>,...,''A''<sub>''g''</sub> and ''B''<sub>1</sub>,...,''B''<sub>''g''</sub> in degree 1, and the class ''P'' of a point in degree 2. The product is given by: ''A''<sub>''i''</sub>''A''<sub>''j''</sub> = ''B''<sub>''i''</sub>''B''<sub>''j''</sub> = 0 for all ''i'' and ''j'', ''A''<sub>''i''</sub>''B''<sub>''j''</sub> = 0 if ''i'' ≠ ''j'', and ''A''<sub>''i''</sub>''B''<sub>''i''</sub> = ''P'' for all ''i''.{{sfn|Hatcher|2001|loc=Example 3.7}} By graded-commutativity, it follows that {{math|1=''B''<sub>''i''</sub>''A''<sub>''i''</sub> = −''P''}}.
*On any topological space, graded-commutativity of the cohomology ring implies that 2''x''<sup>2</sup> = 0 for all odd-degree cohomology classes ''x''. It follows that for a ring ''R'' containing 1/2, all odd-degree elements of ''H''<sup>*</sup>(''X'',''R'') have square zero. On the other hand, odd-degree elements need not have square zero if ''R'' is '''Z'''/2 or '''Z''', as one sees in the example of '''RP'''<sup>2</sup> (with '''Z'''/2 coefficients) or '''RP'''<sup>4</sup> × '''RP'''<sup>2</sup> (with '''Z''' coefficients).

==The diagonal==
The cup product on cohomology can be viewed as coming from the [[diagonal map]] <math>\Delta:X\to X\times X</math>, <math>x\mapsto (x,x)</math>. Namely, for any spaces <math>X</math> and <math>Y</math> with cohomology classes <math>u\in H^{i}(X,R)</math> and <math>v\in H^{j}(Y,R)</math>, there is an '''external product''' (or '''cross product''') cohomology class <math>u\times v\in H^{i+j}(X\times Y,R)</math>. The cup product of classes <math>u\in H^{i}(X,R)</math> and <math>v\in H^{j}(X,R)</math> can be defined as the pullback of the external product by the diagonal:{{sfn|Hatcher|2001|p=186}}
<math display="block">uv=\Delta^*(u\times v)\in H^{i+j}(X,R).</math>

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

==Poincaré duality==
{{Main|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 <math>X</math> be a closed [[connected space|connected]] oriented manifold of dimension <math>n</math>, and let <math>F</math> be a field. Then <math>H^n(X,F)</math> is isomorphic to <math>F</math>, and the product
:<math>H^i(X,F)\times H^{n-i}(X,F)\to H^n(X,F)\cong F</math>
is a [[perfect pairing]] for each integer <math>i</math>.{{sfn|Hatcher|2001|loc=Proposition 3.38}} In particular, the vector spaces <math>H^i(X,F)</math> and <math>H^{n-i}(X,F)</math> have the same (finite) dimension. Likewise, the product on integral cohomology modulo [[torsion subgroup|torsion]] with values in <math>H^n(X,\Z)\cong\Z</math> is a perfect pairing over <math>\Z</math>.

==Characteristic classes==
{{Main|Characteristic class}}
An oriented real [[vector bundle]] ''E'' of rank ''r'' over a topological space ''X'' determines a cohomology class on ''X'', the '''[[Euler class]]''' χ(''E'') ∈ ''H''<sup>''r''</sup>(''X'','''Z'''). Informally, the Euler class is the class of the zero set of a general [[section (fiber bundle)|section]] of ''E''. That interpretation can be made more explicit when ''E'' is a smooth vector bundle over a smooth manifold ''X'', since then a general smooth section of ''X'' vanishes on a codimension-''r'' submanifold of ''X''.

There are several other types of [[characteristic class]]es for vector bundles that take values in cohomology, including [[Chern class]]es, [[Stiefel–Whitney class]]es, and [[Pontryagin class]]es.

==Eilenberg–MacLane spaces==
{{Main|Eilenberg–MacLane space}}
For each abelian group ''A'' and natural number ''j'', there is a space <math>K(A,j)</math> whose ''j''-th homotopy group is isomorphic to ''A'' and whose other homotopy groups are zero. Such a space is called an '''Eilenberg–MacLane space'''. This space has the remarkable property that it is a '''classifying space''' for cohomology: there is a natural element ''u'' of <math>H^j(K(A,j),A)</math>, and every cohomology class of degree ''j'' on every space ''X'' is the pullback of ''u'' by some continuous map <math>X\to K(A,j)</math>. More precisely, pulling back the class ''u'' gives a bijection
:<math>[X, K(A,j)] \stackrel{\cong}{\to} H^j(X,A)</math>
for every space ''X'' with the homotopy type of a CW complex.{{sfn|May|1999|p=177}} Here <math>[X,Y]</math> denotes the set of homotopy classes of continuous maps from ''X'' to ''Y''.

For example, the space <math>K(\Z,1)</math> (defined up to homotopy equivalence) can be taken to be the circle <math>S^1</math>. So the description above says that every element of <math>H^1(X,\Z)</math> is pulled back from the class ''u'' of a point on <math>S^1</math> by some map <math>X\to S^1</math>.

There is a related description of the first cohomology with coefficients in any abelian group ''A'', say for a CW complex ''X''. Namely, <math>H^1(X,A)</math>  is in one-to-one correspondence with the set of isomorphism classes of Galois [[covering space]]s of ''X'' with group ''A'', also called [[principal bundle|principal ''A''-bundles]] over ''X''. For ''X'' connected, it follows that <math>H^1(X,A)</math> is isomorphic to <math>\operatorname{Hom}(\pi_1(X),A)</math>, where <math>\pi_1(X)</math> is the [[fundamental group]] of ''X''. For example, <math>H^1(X,\Z/2)</math> classifies the double covering spaces of ''X'', with the element <math>0\in H^1(X,\Z/2)</math> corresponding to the trivial double covering, the disjoint union of two copies of ''X''.

==Cap product==
{{Main|Cap product}}
For any topological space ''X'', the '''cap product''' is a bilinear map
:<math>\cap: H^i(X,R)\times H_j(X,R) \to H_{j-i}(X,R)</math>
for any integers ''i'' and ''j'' and any commutative ring ''R''. The resulting map
:<math>H^*(X,R)\times H_*(X,R) \to H_*(X,R)</math>
makes the singular homology of ''X'' into a module over the singular cohomology ring of ''X''.

For ''i'' = ''j'', the cap product gives the natural homomorphism
:<math>H^i(X,R)\to \operatorname{Hom}_R(H_i(X,R),R),</math>
which is an isomorphism for ''R'' a field.

For example, let ''X'' be an oriented manifold, not necessarily compact. Then a closed oriented codimension-''i'' submanifold ''Y'' of ''X'' (not necessarily compact) determines an element of ''H''<sup>''i''</sup>(''X'',''R''), and a compact oriented ''j''-dimensional submanifold ''Z'' of ''X'' determines an element of ''H''<sub>''j''</sub>(''X'',''R''). The cap product [''Y''] ∩ [''Z''] ∈ ''H''<sub>''j''−''i''</sub>(''X'',''R'') can be computed by perturbing ''Y'' and ''Z'' to make them intersect transversely and then taking the class of their intersection, which is a compact oriented submanifold of dimension ''j'' − ''i''.

A closed oriented manifold ''X'' of dimension ''n'' has a [[fundamental class]] [''X''] in ''H''<sub>''n''</sub>(''X'',''R''). The Poincaré duality isomorphism
<math display="block">H^i(X,R)\overset{\cong}{\to} H_{n-i}(X,R)</math>
is defined by cap product with the fundamental class of ''X''.

==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 of ''dual cell structure'', which [[Henri Poincaré]] used 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.{{sfn|Dieudonné|1989|loc=Section IV.3}} In the mid-1920s, [[James Waddell Alexander II|J. W. Alexander]] and [[Solomon Lefschetz]] founded [[intersection theory]] of cycles on manifolds.  On a closed oriented ''n''-dimensional manifold ''M'' an ''i''-cycle and a ''j''-cycle with nonempty intersection will, if in the [[general position]], have as their intersection a (''i''&nbsp;+&nbsp;''j''&nbsp;&minus;&nbsp;''n'')-cycle. This leads to a multiplication of homology classes
:<math>H_i(M) \times H_j(M) \to H_{i+j-n}(M),</math>
which (in retrospect) can be identified with the [[cup product]] on the cohomology of ''M''.

Alexander had by 1930 defined a first notion of a cochain, by thinking of an ''i''-cochain on a space ''X'' as a function on small neighborhoods of the diagonal in ''X''<sup>''i''+1</sup>.

In 1931, [[Georges de Rham]] related homology and differential forms, proving [[De Rham cohomology#De Rham's theorem|de Rham's theorem]]. This result can be stated more simply in terms of cohomology.

In 1934, [[Lev Pontryagin]] proved the [[Pontryagin duality]] theorem; a result on [[topological group]]s.  This (in rather special cases) provided an interpretation of Poincaré duality and [[Alexander duality]] in terms of group [[character (mathematics)|character]]s.

At a 1935 conference in [[Moscow]], [[Andrey Kolmogorov]] and Alexander both introduced cohomology and tried to construct a cohomology product structure.

In 1936, [[Norman Steenrod]] constructed [[Čech cohomology]] by dualizing Čech homology.

From 1936 to 1938, [[Hassler Whitney]] and [[Eduard Čech]] developed the [[cup product]] (making cohomology into a graded ring) and [[cap 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 Eilenberg]] overcame the technical limitations, and gave the modern definition of singular homology and cohomology.

In 1945, Eilenberg and Steenrod stated the [[Eilenberg–Steenrod axioms|axioms]] defining 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 Leray]] defined sheaf cohomology.

In 1948 [[Edwin Spanier]], building on work of Alexander and Kolmogorov, developed [[Alexander–Spanier cohomology]].

==Sheaf cohomology==
{{Main|Sheaf cohomology}}
'''Sheaf cohomology''' is a rich generalization of singular cohomology, allowing more general "coefficients" than simply an abelian group. For every [[sheaf (mathematics)|sheaf]] of abelian groups ''E'' on a topological space ''X'', one has cohomology groups ''H''<sup>''i''</sup>(''X'',''E'') for integers ''i''. In particular, in the case of the [[constant sheaf]] on ''X'' associated with an abelian group ''A'', the resulting groups ''H''<sup>''i''</sup>(''X'',''A'') coincide with singular cohomology for ''X'' a manifold or CW complex (though not for arbitrary spaces ''X''). Starting in the 1950s, sheaf cohomology has become a central part of [[algebraic geometry]] and [[complex analysis]], partly because of the importance of the sheaf of [[regular function]]s or the sheaf of [[holomorphic function]]s.

[[Alexander Grothendieck|Grothendieck]] elegantly defined and characterized sheaf cohomology in the language of [[homological algebra]]. The essential point is to fix the space ''X'' and think of sheaf cohomology as a functor from the [[abelian category]] of sheaves on ''X'' to abelian groups. Start with the functor taking a sheaf ''E'' on ''X'' to its abelian group of global sections over ''X'', ''E''(''X''). This functor is [[left exact functor|left exact]], but not necessarily right exact. Grothendieck defined sheaf cohomology groups to be the right [[derived functor]]s of the left exact functor ''E'' ↦ ''E''(''X'').{{sfn|Hartshorne|1977|loc=Section III.2}}

That definition suggests various generalizations. For example, one can define the cohomology of a topological space ''X'' with coefficients in any complex of sheaves, earlier called [[hypercohomology]] (but usually now just "cohomology"). From that point of view, sheaf cohomology becomes a sequence of functors from the [[derived category]] of sheaves on ''X'' to 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 ring ''R'', the [[Tor functor|Tor group]]s Tor<sub>''i''</sub><sup>''R''</sup>(''M'',''N'') form a "homology theory" in each variable, the left derived functors of the tensor product ''M''⊗<sub>''R''</sub>''N'' of ''R''-modules. Likewise, the [[Ext group]]s Ext<sup>''i''</sup><sub>''R''</sub>(''M'',''N'') can be viewed as a "cohomology theory" in each variable, the right derived functors of the Hom functor Hom<sub>''R''</sub>(''M'',''N'').

Sheaf cohomology can be identified with a type of Ext group. Namely, for a sheaf ''E'' on a topological space ''X'', ''H''<sup>''i''</sup>(''X'',''E'') is isomorphic to Ext<sup>''i''</sup>('''Z'''<sub>''X''</sub>, ''E''), where '''Z'''<sub>''X''</sub> denotes the constant sheaf associated with the integers '''Z''', and Ext is taken in the abelian category of sheaves on ''X''.

== Cohomology of varieties ==
There are numerous machines built for computing the cohomology of [[algebraic varieties]]. The simplest case being the determination of cohomology for [[smooth variety|smooth]] [[projective varieties]] over a field of [[characteristic (algebra)|characteristic]] <math>0</math>. Tools from [[Hodge theory]], called [[Hodge structure]]s, 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 <math>\mathbb{P}^n</math> can be determined from the degree of the polynomial alone.

When considering varieties over a [[finite field]], or a field of characteristic <math>p</math>, 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 [[étale topology]] to define the cohomology theory for varieties over a finite field. Using the étale topology for a variety over a field of characteristic <math>p</math> one can construct [[l-adic cohomology|<math>\ell</math>-adic cohomology]] for <math>\ell\neq p</math>. This is defined as the [[projective limit]]

:<math>H^k(X;\Q_\ell) := \varprojlim_{n\in\mathbb N} H^k_{et}(X;\Z/(\ell^n)) \otimes_{\Z_\ell} \Q_\ell.</math>

If we have a scheme of finite type

:<math>X = \operatorname{Proj} \left( \frac{\Z \left[x_0,\ldots,x_n \right]}{ \left (f_1,\ldots,f_k \right )} \right)</math>

then there is an equality of dimensions for the Betti cohomology of <math>X(\Complex)</math> and the <math>\ell</math>-adic cohomology of <math>X(\mathbb{F}_q)</math> whenever the variety is smooth over both fields. In addition to these cohomology theories there are other cohomology theories called [[Weil cohomology theory|Weil cohomology theories]] which behave similarly to singular cohomology. There is a conjectured theory of motives which underlie all of the Weil cohomology theories.
<!-- Discuss lefschetz fixed point theorem and weil conjectures -->

Another useful computational tool is the blowup sequence. Given a codimension <math>\geq 2</math> subscheme <math>Z \subset X</math> there is a [[Cartesian square (category theory)|Cartesian square]]

:<math>\begin{matrix}
E & \longrightarrow & Bl_Z(X) \\
\downarrow & & \downarrow \\
Z & \longrightarrow & X
\end{matrix}</math>

From this there is an associated long exact sequence

:<math>\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</math>

If the subvariety <math>Z</math> is smooth, then the connecting morphisms are all trivial, hence

:<math>H^n(Bl_Z(X))\oplus H^n(Z) \cong H^n(X) \oplus H^n(E)</math>
<!--In addition, the cohomology ring of the blowup can be readily computed using the chern classes of the normal bundle <math>N_{Z/X}</math>. It is given by
:<math>
H^*()
</math>
-->

==Axioms and generalized cohomology theories==
{{anchor|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 cohomology]] or [[sheaf 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 the [[Eilenberg–Steenrod axioms]], and any two constructions that share those properties will agree at least on all CW complexes.{{sfn|May|1999|p=95}} 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 [[simplicial cohomology]] for [[simplicial complex]]es, [[cellular homology|cellular cohomology]] for CW complexes, and [[de Rham cohomology]] for smooth manifolds.

One of the Eilenberg–Steenrod axioms for a cohomology theory is the '''dimension axiom''': if ''P'' is a single point, then ''H<sup>i</sup>''(''P'') = 0 for all ''i'' ≠ 0. Around 1960, [[George W. Whitehead]] 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 space, not directly accessible from singular cohomology. (In this context, singular cohomology is often called "ordinary cohomology".)

By definition, a '''generalized homology theory''' is a sequence of [[functor]]s ''h''<sub>''i''</sub> (for integers ''i'') from the [[category (mathematics)|category]] of CW-[[topological pair|pairs]] (''X'',&nbsp;''A'') (so ''X'' is a CW complex and ''A'' is a subcomplex) to the category of abelian groups, together with a [[natural transformation]] {{math|∂<sub>''i''</sub>: ''h''<sub>''i''</sub>(''X'', ''A'') → ''h''<sub>''i''−1</sub>(''A'')}} called the '''boundary homomorphism''' (here ''h''<sub>''i''−1</sub>(''A'') is a shorthand for ''h''<sub>''i''−1</sub>(''A'',∅)). The axioms are:

# '''Homotopy''': If <math>f:(X,A) \to (Y,B)</math> is homotopic to <math>g: (X,A) \to (Y,B)</math>, then the induced homomorphisms on homology are the same.
# '''Exactness''': Each pair (''X'',''A'') induces a long exact sequence in homology, via the inclusions {{math|''f'': ''A'' → ''X''}} and {{math|''g'': (''X'',∅) → (''X'',''A'')}}: <math display="block"> \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.</math>
# '''[[Excision theorem|Excision]]''': If ''X'' is the union of subcomplexes ''A'' and ''B'', then the inclusion ''f'': (''A'',''A''∩''B'') → (''X'',''B'') induces an isomorphism <math display="block"> h_i(A, A\cap B) \overset{f_*}{\to} h_i(X,B)</math> for every ''i''.
# '''Additivity''': If (''X'',''A'') is the disjoint union of a set of pairs (''X''<sub>''α''</sub>,''A''<sub>''α''</sub>), then the inclusions (''X''<sub>''α''</sub>,''A''<sub>''α''</sub>) → (''X'',''A'') induce an isomorphism from the [[Direct sum of modules#Construction for an arbitrary family of modules|direct sum]]: <math display="block"> \bigoplus_{\alpha} h_i(X_\alpha,A_\alpha)\to h_i(X,A)</math> for every ''i''.

The axioms for a generalized cohomology theory are obtained by reversing the arrows, roughly speaking. In more detail, a '''generalized cohomology theory''' is a sequence of contravariant functors ''h''<sup>''i''</sup> (for integers ''i'') from the category of CW-pairs to the category of abelian groups, together with a natural transformation {{math|''d'': ''h''<sup>''i''</sup>(''A'') → ''h''<sup>''i''+1</sup>(''X'',''A'')}} called the '''boundary homomorphism''' (writing ''h''<sup>''i''</sup>(''A'') for ''h''<sup>''i''</sup>(''A'',∅)). The axioms are:

# '''Homotopy''': Homotopic maps induce the same homomorphism on cohomology.
# '''Exactness''': Each pair (''X'',''A'') induces a long exact sequence in cohomology, via the inclusions ''f'': ''A'' → ''X'' and ''g'': (''X'',∅) → (''X'',''A''): <math display="block"> \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.</math>
# '''Excision''': If ''X'' is the union of subcomplexes ''A'' and ''B'', then the inclusion ''f'': (''A'',''A''∩''B'') → (''X'',''B'') induces an isomorphism <math display="block"> h^i(X,B) \overset{f_*}{\to} h^i(A,A\cap B)</math> for every ''i''.
# '''Additivity''': If (''X'',''A'') is the disjoint union of a set of pairs (''X''<sub>''α''</sub>,''A''<sub>''α''</sub>), then the inclusions (''X''<sub>''α''</sub>,''A''<sub>''α''</sub>) → (''X'',''A'') induce an isomorphism to the [[Direct product of groups#Infinite direct products|product group]]: <math display="block"> h^i(X,A)\to \prod_\alpha h^i(X_\alpha,A_\alpha)</math> for every ''i''.

A [[spectrum (topology)|spectrum]] determines both a generalized homology theory and a generalized cohomology theory. A fundamental result by Brown, Whitehead, and [[Frank Adams|Adams]] says that every generalized homology theory comes from a spectrum, and likewise every generalized cohomology theory comes from a spectrum.{{sfn|Switzer|1975|loc=Theorem 9.27; Corollary 14.36; Remarks|p=117, 331}} 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 (called [[phantom map]]s) 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.<ref>{{cite web|url=https://mathoverflow.net/q/117684 |title=Are spectra really the same as cohomology theories?|website=MathOverflow}}</ref> It is the stable homotopy category, not these other categories, that has good properties such as being [[triangulated category|triangulated]].

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 [[weak homotopy equivalence]] 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 corresponding theory on CW complexes.{{sfn|Switzer|1975|loc=7.68}}

Some examples of generalized cohomology theories are:
* Stable [[cohomotopy group]]s <math>\pi_S^*(X).</math> The corresponding homology theory is used more often: [[stable homotopy theory|stable homotopy groups]] <math>\pi^S_*(X).</math>
* Various different flavors of [[cobordism]] groups, based on studying a space by considering all maps from it to manifolds: unoriented cobordism <math>MO^*(X)</math> oriented cobordism <math>MSO^*(X),</math> [[complex cobordism]] <math>MU^*(X),</math> and so on. Complex cobordism has turned out to be especially powerful in homotopy theory. It is closely related to [[formal group]]s, via a theorem of [[Daniel Quillen]].
* Various different flavors of topological [[K-theory]], based on studying a space by considering all vector bundles over it: <math>KO^*(X)</math> (real periodic K-theory), <math>ko^*(X)</math> (real connective K-theory), <math>K^*(X)</math> (complex periodic K-theory), <math>ku^*(X)</math> (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 of [[elliptic cohomology]].
Many of these theories carry richer information than ordinary cohomology, but are harder to compute.

A cohomology theory ''E'' is said to be '''multiplicative''' if <math>E^*(X)</math> has the structure of a graded ring for each space ''X''. In the language of spectra, there are several more precise notions of a [[ring spectrum]], such as an [[highly structured ring spectrum|''E''<sub>∞</sub> 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:
{{div col}}
*[[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 group]]s
*[[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]]
{{div col end}}

== See also ==
*[[complex-oriented cohomology theory]]<!-- should be mentioned somehow -->

==Citations==
{{Reflist|30em}}

== References ==
*{{Citation | author1-first=Jean | author1-last=Dieudonné | author1-link=Jean Dieudonné | title=History of Algebraic and Differential Topology | publisher=[[Birkhäuser]] | year=1989 | mr=0995842 | isbn=0-8176-3388-X | url-access=registration | url=https://archive.org/details/historyofalgebra0000dieu_g9a3 }}
*{{Citation | author1-first=Albrecht | author1-last=Dold | author1-link=Albrecht Dold | title=Lectures on Algebraic Topology | publisher=[[Springer-Verlag]] | year=1972 | mr=0415602 | isbn=978-3-540-58660-9}}
*{{Citation | author1-first=Samuel | author1-last=Eilenberg | author1-link=Samuel Eilenberg | author2-first=Norman | author2-last=Steenrod | author2-link=Norman Steenrod | title=Foundations of Algebraic Topology | publisher=[[Princeton University Press]] | year=1952 | mr=0050886 | isbn=9780691627236}}
*{{Citation | author1-last=Hartshorne | author1-first=Robin | author1-link=Robin Hartshorne | title=Algebraic Geometry | publisher=[[Springer-Verlag]] | location=New York, Heidelberg | series=Graduate Texts in Mathematics | isbn=0-387-90244-9 | mr=0463157 | year=1977 | volume=52}}
*{{Citation | author1-first=Allen | author1-last=Hatcher | author1-link=Allen Hatcher | title=Algebraic Topology | publisher=[[Cambridge University Press]] | year=2001 | url=http://pi.math.cornell.edu/~hatcher/AT/ATpage.html | isbn=0-521-79540-0 | mr =1867354}}
*{{Springer |title=Cohomology |id=p/c023060 }}.
*{{Citation | author1-first=J. Peter | author1-last=May | author1-link=J. Peter May | title=A Concise Course in Algebraic Topology | publisher=[[University of Chicago Press]] | year=1999 | url=http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf | mr=1702278 | isbn=0-226-51182-0}}
*{{Citation | author1-first=Robert | author1-last=Switzer | title=Algebraic Topology — Homology and Homotopy | publisher=[[Springer-Verlag]] | year=1975 | mr=0385836 | isbn=3-540-42750-3}}
*{{Citation |author1-first=René |author1-last=Thom |author1-link=René Thom |mr=0061823 |title=Quelques propriétés globales des variétés différentiables |journal=[[Commentarii Mathematici Helvetici]] |volume=28 |year=1954 |pages=17–86 |url=http://eudml.org/doc/139072 |doi=10.1007/BF02566923 |s2cid=120243638 }}

{{Topology}}
{{Authority control}}

[[Category:Cohomology theories|*]]