Cohomology

2025-11-03T22:03:06Z

2600:4040:2C98:EF00:C01:C97A:36EF:D08B:

{{Short description|Algebraic structure used in topology}}
{{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''' 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>th '''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''''i''''X'' ≅ ''H''''n''−''i''''X''. As a result, a closed oriented submanifold ''S'' of [[codimension]] ''i'' in ''X'' determines a cohomology class in ''H''''i''''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)|transversally]], 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 transversally, 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'''''n''.
*[[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''2) (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''1)2 has a basis as a [[free module|free '''Z'''-module]] of the form: the element 1 in degree 0, ''x'' := [''P'' × ''S''1] and ''y'' := [''S''1 × ''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''*(''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'''''n'' with '''Z'''/2 coefficients is '''Z'''/2[''x'']/(''x''''n''+1), with ''x'' in degree 1.{{sfn|Hatcher|2001|loc=Theorem 3.19}} Here ''x'' is the class of a [[hyperplane]] '''RP'''''n''−1 in '''RP'''''n''; this makes sense even though '''RP'''''j'' 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'''2''a'' 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''''i'' for ''i''=1,...,''a''. The '''Z'''-cohomology of '''RP'''2''a''+1 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'''''n'' is '''Z'''[''x'']/(''x''''n''+1), with ''x'' in degree 2.{{sfn|Hatcher|2001|loc=Theorem 3.19}} Here ''x'' is the class of a hyperplane '''CP'''''n''−1 in '''CP'''''n''. More generally, ''x''''j'' is the class of a linear subspace '''CP'''''n''−''j'' in '''CP'''''n''.
*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''1,...,''A''''g'' and ''B''1,...,''B''''g'' in degree 1, and the class ''P'' of a point in degree 2. The product is given by: ''A''''i''''A''''j'' = ''B''''i''''B''''j'' = 0 for all ''i'' and ''j'', ''A''''i''''B''''j'' = 0 if ''i'' ≠ ''j'', and ''A''''i''''B''''i'' = ''P'' for all ''i''.{{sfn|Hatcher|2001|loc=Example 3.7}} By graded-commutativity, it follows that {{math|1=''B''''i''''A''''i'' = −''P''}}.
*On any topological space, graded-commutativity of the cohomology ring implies that 2''x''2 = 0 for all odd-degree cohomology classes ''x''. It follows that for a ring ''R'' containing 1/2, all odd-degree elements of ''H''*(''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'''2 (with '''Z'''/2 coefficients) or '''RP'''4 × '''RP'''2 (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''''r''(''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''''i''(''X'',''R''), and a compact oriented ''j''-dimensional submanifold ''Z'' of ''X'' determines an element of ''H''''j''(''X'',''R''). The cap product [''Y''] ∩ [''Z''] ∈ ''H''''j''−''i''(''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''''n''(''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'' + ''j'' − ''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''''i''+1.

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''''i''(''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''''i''(''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''i''''R''(''M'',''N'') form a "homology theory" in each variable, the left derived functors of the tensor product ''M''⊗''R''''N'' of ''R''-modules. Likewise, the [[Ext group]]s Ext''i''''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 sheaf ''E'' on a topological space ''X'', ''H''''i''(''X'',''E'') is isomorphic to Ext''i''('''Z'''''X'', ''E''), where '''Z'''''X'' 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.


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>


==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 ''Hi''(''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''''i'' (for integers ''i'') from the [[category (mathematics)|category]] of CW-[[topological pair|pairs]] (''X'', ''A'') (so ''X'' is a CW complex and ''A'' is a subcomplex) to the category of abelian groups, together with a [[natural transformation]] {{math|∂''i'': ''h''''i''(''X'', ''A'') → ''h''''i''−1(''A'')}} called the '''boundary homomorphism''' (here ''h''''i''−1(''A'') is a shorthand for ''h''''i''−1(''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''''α'',''A''''α''), then the inclusions (''X''''α'',''A''''α'') → (''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''''i'' (for integers ''i'') from the category of CW-pairs to the category of abelian groups, together with a natural transformation {{math|''d'': ''h''''i''(''A'') → ''h''''i''+1(''X'',''A'')}} called the '''boundary homomorphism''' (writing ''h''''i''(''A'') for ''h''''i''(''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''''α'',''A''''α''), then the inclusions (''X''''α'',''A''''α'') → (''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''∞ 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]]

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

== References ==
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