Cohomology theory

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, specifically in
homology theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
and
algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...
, cohomology is a general term for a sequence of
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
s, usually one associated with a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, often defined from a
cochain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemat ...
. 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 Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
s on the group of
chains A chain is a assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a in that it is flexible and d in but , rigid, and load-bearing in . A chain may consist of two or more links. C ...
in homology theory. From its beginning in
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

, 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 Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ...

and
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...
. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do with functions and
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a Pushforward (disambiguation), pushforward. Precomposition Precomposition with a Function (mathematics), function probabl ...
s in geometric situations: given spaces ''X'' and ''Y'', and some kind of function ''F'' on ''Y'', for any
mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated ...
, composition with ''f'' gives rise to a function on ''X''. The most important cohomology theories have a product, the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutati ...
, which gives them a
ring Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
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 In Abstract algebra, algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements ''x'', ''y'' satisfy :xy = (-1)^ yx , where , denote the degrees o ...
with any topological space. Every
continuous map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
''f'': ''X'' → ''Y'' determines a
homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
from 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 as
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about Loop (topology), loops in a Mathematical space, space. I ...
s, 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 the
singular chain complex In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-d ...
: : $\cdots \to C_\stackrel C_i \stackrel\ C_ \to \cdots$ By definition, the
singular homology In algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify ...
of ''X'' is the homology of this chain complex (the kernel of one homomorphism modulo the image of the previous one). In more detail, ''Ci'' is the
free abelian group In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation that is associative, commutative, and invertible. A basis, also called ...
on the set of continuous maps from the standard ''i''-simplex to ''X'' (called "singular ''i''-simplices in ''X''"), and ∂''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 ''Ci'' by its dual group $C_i^* := \mathrm\left(C_i,A\right),$ and $\partial_i$ by its dual homomorphism : $d_: C_^* \to C_^*.$ This has the effect of "reversing all the arrows" of the original complex, leaving a
cochain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathemat ...
: $\cdots \leftarrow C_^* \stackrel\ C_^* \stackrel C_^* \leftarrow \cdots$ For an integer ''i'', the ''i''th cohomology group of ''X'' with coefficients in ''A'' is defined to be ker(''di'')/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 called singular ''i''-cochains with 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 im(''d'') are called cocycles and coboundaries, respectively, while elements of ker(''d'')/im(''d'') = ''H''''i''(''X'', ''A'') are called cohomology classes (because they are
equivalence class In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
es of cocycles). In what follows, the coefficient group ''A'' is sometimes not written. It is common to take ''A'' to be a
commutative ring In ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical ana ...
''R''; then the cohomology groups are ''R''-
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modula ...
s. A standard choice is the ring Z of
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s. Some of the formal properties of cohomology are only minor variants of the properties of homology: * A continuous map $f: X \to Y$ determines a pushforward homomorphism $f_*:H_i\left(X\right) \to H_i\left(Y\right)$ on homology and a pullback homomorphism $f^*: H^i\left(Y\right) \to H^i\left(X\right)$ on cohomology. This makes cohomology into a
contravariant functor In mathematics, specifically category theory, a functor is a wikt:mapping, mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are ass ...
from topological spaces to abelian groups (or ''R''-modules). * Two
homotopic In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric o ...
maps from ''X'' to ''Y'' induce the same homomorphism on cohomology (just as on homology). * The
Mayer–Vietoris sequence In mathematics, particularly algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces, known as their Homology group, homology and cohomology groups. The r ...
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 ''X'' is the union of
open subset Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australia ...
s ''U'' and ''V'', then there is a
long exact sequence An exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Image (mathematics ...
: ::$\cdots \to H^i\left(X\right) \to H^i\left(U\right)\oplus H^i\left(V\right) \to H^i\left(U\cap V\right) \to H^\left(X\right) \to \cdots$ * There are relative cohomology groups $H^i\left(X,Y;A\right)$ for any ''Y'' of a space ''X''. They are related to the usual cohomology groups by a long exact sequence: ::$\cdots \to H^i\left(X,Y\right) \to H^i\left(X\right) \to H^i\left(Y\right) \to H^\left(X,Y\right) \to \cdots$ * The
universal coefficient theorem In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'': : completely ...
describes cohomology in terms of homology, using
Ext groupIn mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic struct ...
s. Namely, there is a
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Image (mathematics ...
::$0 \to \operatorname_^1\left(\operatorname_\left(X, \Z\right), A\right) \to H^i\left(X, A\right) \to \operatorname_\left(H_i\left(X,\Z\right), A\right)\to 0.$ :A related statement is that for a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
''F'', $H^i\left(X,F\right)$ is precisely the
dual space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
of the
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
$H_i\left(X,F\right)$. * If ''X'' is a topological
manifold In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

or a
CW complex A CW complex is a kind of a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantiti ...
, then the cohomology groups $H^i\left(X,A\right)$ are zero for ''i'' greater than the
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

of ''X''. If ''X'' is a compact manifold (possibly with boundary), or a CW complex with finitely many cells in each dimension, and ''R'' is a commutative
Noetherian ring In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, then the ''R''-module ''H''''i''(''X'',''R'') is finitely generated for 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 a
bilinear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
, called the
cup product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutati ...
: :$H^i\left(X,R\right)\times H^j\left(X,R\right) \to H^\left(X,R\right),$ defined by an explicit formula on singular cochains. The product of cohomology classes ''u'' and ''v'' is written as ''u'' ∪ ''v'' or simply as ''uv''. This product makes the
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
:$H^*\left(X,R\right)=\bigoplus_i H^i\left(X,R\right)$ into a
graded ring In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, called the
cohomology ring In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
of ''X''. It is
graded-commutative In Abstract algebra, algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements ''x'', ''y'' satisfy :xy = (-1)^ yx , where , denote the degrees o ...
in the sense that: :$uv=\left(-1\right)^vu, \qquad u \in H^i\left(X,R\right), v \in H^j\left(X,R\right).$ For any continuous map $f\colon X\to Y,$ the
pullback In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
$f^*\colon H^*\left(Y,R\right) \to H^*\left(X, R\right)$ is a homomorphism of graded ''R''-
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the broad areas of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), ...
s. It follows that if two spaces are
homotopy equivalent In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry) ...
, then their cohomology rings are isomorphic. Here are some of the geometric interpretations of the cup product. In what follows, manifolds are understood to be without boundary, unless stated otherwise. A 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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
of ''M'', not necessarily compact (although ''N'' is automatically compact if ''M'' is). * Let ''X'' be a closed
oriented In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
''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 transversely, then :: :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 In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
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. :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. :Note that Thom constructed an integral cohomology class of degree 7 on a smooth 14-manifold that is not the class of any smooth submanifold. On the other hand, he showed that every integral cohomology class of positive degree on a smooth manifold has a positive multiple that is the class of a smooth submanifold. 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
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coefficients is isomorphic to the de Rham cohomology of ''X'', defined using
differential form In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
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 homological algebra in mathematics, the homotopy category ''K(A)'' of chain complexes in an additive category ''A'' is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain co ...

. In fact, it is impossible to modify the definition of singular cochains with coefficients in the integers $\Z$ or in $\Z/p$ 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 In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology. For a given prime number p, the Steenrod algebra A_p is the graded Hopf algebra over the field \mathbb_p of order p, c ...
s on mod ''p'' cohomology. Very informally, for any topological space ''X'', elements of $H^i\left(X\right)$ 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 $H^i\left(X\right)$ 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 as lying on the subspace $f^\left(N\right)$ of ''X''; this is justified in that the class restricts to zero in the cohomology of the open subset $X-f^\left(N\right).$ The cohomology class 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 In mathematics, a topological space ''X'' is contractible if the identity function, identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to ...
space, such as Euclidean space R''n''. *For a positive integer ''n'', the cohomology ring of the
sphere A sphere (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...
$S^n$ is Z 'x''(''x''2) (the
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space (linear algebra), quot ...
of a
polynomial ring In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
by the given 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 In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative posit ...

$\left(S^1\right)^n$ is the
exterior algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
over Z on ''n'' generators in degree 1. For example, let ''P'' denote a point in the circle $S^1$, and ''Q'' the point (''P'',''P'') in the 2-dimensional torus $\left(S^1\right)^2$. Then the cohomology of (''S''1)2 has a basis as a of the form: the element 1 in degree 0, ''x'' := 'P'' × ''S''1and ''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 Künneth is a surname. Notable people with the surname include: * Hermann Künneth Hermann Lorenz Künneth (July 6, 1892 Neustadt an der Haardt – May 7, 1975 Erlangen Erlangen (; East Franconian: ''Erlang'', Bavarian: ''Erlanga'') is a Mid ...
gives that the cohomology ring of the
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''X'' × ''Y'' is a
tensor product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of ''R''-algebras: ::$H^*\left(X\times Y,R\right)\cong H^*\left(X,R\right)\otimes_R H^*\left(Y,R\right).$ * The cohomology ring of
real projective space In mathematics, real projective space, or RP''n'' or \mathbb_n(\mathbb), is the topological space of lines passing through the origin 0 in R''n''+1. It is a compact space, compact, smooth manifold of dimension ''n'', and is a special case Gr(1, R''n ...
RP''n'' with Z/2 coefficients is Z/2 'x''(''x''''n''+1), with ''x'' in degree 1. Here ''x'' is the class of a
hyperplane In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position ...
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. :With integer coefficients, the answer is a bit more complicated. The Z-cohomology of RP2''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 RP2''a''+1 is the same together with an extra copy of Z in degree 2''a''+1. *The cohomology ring of
complex projective space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
CP''n'' is Z 'x''(''x''''n''+1), with ''x'' in degree 2. 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 Genus /ˈdʒiː.nəs/ (plural genera /ˈdʒen.ər.ə/) is a taxonomic rank In biological classification In biology, taxonomy () is the scientific study of naming, defining (Circumscription (taxonomy), circumscribing) and classifying gr ...
''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''. By graded-commutativity, it follows that ''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 RP2 (with Z/2 coefficients) or RP4 × RP2 (with Z coefficients).

# The diagonal

The cup product on cohomology can be viewed as coming from the
diagonal map In geometry, a diagonal is a line segment joining two vertex (geometry), vertices of a polygon or polyhedron, when those vertices are not on the same edge (geometry), edge. Informally, any sloping line is called diagonal. The word ''diagonal'' ...
Δ: ''X'' → ''X'' × ''X'', ''x'' ↦ (''x'',''x''). Namely, for any spaces ''X'' and ''Y'' with cohomology classes ''u'' ∈ ''H''''i''(''X'',''R'') and ''v'' ∈ ''H''''j''(''Y'',''R''), there is an external product (or cross product) cohomology class ''u'' × ''v'' ∈ ''H''''i''+''j''(''X'' × ''Y'',''R''). The cup product of classes ''u'' ∈ ''H''''i''(''X'',''R'') and ''v'' ∈ ''H''''j''(''X'',''R'') can be defined as the pullback of the external product by the diagonal: :$uv=\Delta^*\left(u\times v\right)\in H^\left(X,R\right).$ Alternatively, the external product can be defined in terms of the cup product. For spaces ''X'' and ''Y'', write ''f'': ''X'' × ''Y'' → ''X'' and ''g'': ''X'' × ''Y'' → ''Y'' for the two projections. Then the external product of classes ''u'' ∈ ''H''''i''(''X'',''R'') and ''v'' ∈ ''H''''j''(''Y'',''R'') is: :$u\times v=\left(f^*\left(u\right)\right)\left(g^*\left(v\right)\right)\in H^\left(X\times Y,R\right).$

# 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 closed
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oriented manifold of dimension ''n'', and let ''F'' be a field. Then ''H''''n''(''X'',''F'') is isomorphic to ''F'', and the product :$H^i\left(X,F\right)\times H^\left(X,F\right)\to H^n\left(X,F\right)\cong F$ is a perfect pairing for 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 modulo torsion subgroup, torsion with values in ''H''''n''(''X'',Z) ≅ Z is a perfect pairing over Z.

# Characteristic classes

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 classes for vector bundles that take values in cohomology, including Chern classes, Stiefel–Whitney classes, and Pontryagin classes.

# Eilenberg–MacLane spaces

For each abelian group ''A'' and natural number ''j'', there is a space $K\left(A,j\right)$ 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 $H^j\left(K\left(A,j\right),A\right)$, and every cohomology class of degree ''j'' on every space ''X'' is the pullback of ''u'' by some continuous map $X\to K\left(A,j\right)$. More precisely, pulling back the class ''u'' gives a bijection :$\left[X, K\left(A,j\right)\right] \stackrel H^j\left(X,A\right)$ for every space ''X'' with the homotopy type of a CW complex. Here $\left[X,Y\right]$ denotes the set of homotopy classes of continuous maps from ''X'' to ''Y''. For example, the space $K\left(\Z,1\right)$ (defined up to homotopy equivalence) can be taken to be the circle $S^1$. So the description above says that every element of $H^1\left(X,\Z\right)$ is pulled back from the class ''u'' of 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 group ''A'', say for a CW complex ''X''. Namely, $H^1\left(X,A\right)$ is in one-to-one correspondence with the set of isomorphism classes of Galois covering spaces of ''X'' with group ''A'', also called principal bundle, principal ''A''-bundles over ''X''. For ''X'' connected, it follows that $H^1\left(X,A\right)$ is isomorphic to $\operatorname\left(\pi_1\left(X\right),A\right)$, where $\pi_1\left(X\right)$ is the fundamental group of ''X''. For example, $H^1\left(X,\Z/2\right)$ classifies the double covering spaces of ''X'', with the element $0\in H^1\left(X,\Z/2\right)$ corresponding to the trivial double covering, the disjoint union of two copies of ''X''.

# Cap product

For any topological space ''X'', the cap product is a bilinear map :$\cap: H^i\left(X,R\right)\times H_j\left(X,R\right) \to H_\left(X,R\right)$ for any integers ''i'' and ''j'' and any commutative ring ''R''. The resulting map :$H^*\left(X,R\right)\times H_*\left(X,R\right) \to H_*\left(X,R\right)$ 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 :$H^i\left(X,R\right)\to \operatorname_R\left(H_i\left(X,R\right),R\right),$ 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 :$H^i\left(X,R\right)\stackrel H_\left(X,R\right)$ is defined by cap product with the fundamental class of ''X''.

# History, to the birth 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 germ of the idea of cohomology, but this was not seen until later. There were various precursors to cohomology. In the mid-1920s, James Waddell Alexander II, J. W. Alexander and Solomon Lefschetz founded the 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 general position, have intersection an (''i'' + ''j'' − ''n'')-cycle. This leads to a multiplication of homology classes :$H_i\left(M\right) \times H_j\left(M\right) \to H_\left(M\right),$ 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'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 groups. This (in rather special cases) provided an interpretation of Poincaré duality and Alexander duality in terms of group character (mathematics), characters. 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

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 functions or the sheaf of holomorphic functions. 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 functors of the left exact functor ''E'' ↦ ''E''(''X''). 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 groups 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 groupIn mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic struct ...
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 projective varieties over a field of characteristic $0$. Tools from Hodge theory, called Hodge 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 smooth hypersurface in $\mathbb^n$ can be determined from the degree of the polynomial alone. When considering varieties over a finite 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 a Grothendieck topology and used sheaf cohomology over the etale topology to 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 cohomology for $\ell\neq p$. This is defined as :$H^k\left(X;\Q_\ell\right) := \varprojlim H^k_\left(X;\Z/\left(\ell^n\right)\right) \otimes_ \Q_\ell$ If we have a scheme of finite type :$X = \text \left\left( \frac \right\right)$ then there is an equality of dimensions for the Betti cohomology of $X\left(\Complex\right)$ and the $\ell$-adic cohomology of $X\left(\mathbb_q\right)$ 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 $\geq 2$ subscheme $Z \subset X$ there is a Cartesian square :$\begin E & \longrightarrow & Bl_Z\left(X\right) \\ \downarrow & & \downarrow \\ Z & \longrightarrow & X \end$ From this there is an associated long exact sequence :$\cdots \to H^n\left(X\right) \to H^n\left(Z\right) \oplus H^n\left(Bl_Z\left(X\right)\right) \to H^n\left(E\right) \to H^\left(X\right) \to \cdots$ If the subvariety $Z$ is smooth, then the connecting morphisms are all trivial, hence :$H^n\left(Bl_Z\left(X\right)\right)\oplus H^n\left(Z\right) \cong H^n\left(X\right) \oplus H^n\left(E\right)$

# Axioms and generalized 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. 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 complexes, 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 functors ''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 ∂''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 $f:\left(X,A\right) \to \left(Y,B\right)$ is homotopic to $g: \left(X,A\right) \to \left(Y,B\right)$, then the induced homomorphisms on homology are the same. # Exactness: Each pair (''X'',''A'') induces a long exact sequence in homology, via the inclusions ''f'': ''A'' → ''X'' and ''g'': (''X'',∅) → (''X'',''A''):$\quad \cdots \to h_i\left(A\right) \overset h_i\left(X\right) \overset h_i \left(X,A\right) \overset h_\left(A\right) \to \cdots.$ # 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 $\quad h_i\left(A, A\cap B\right) \overset h_i\left(X,B\right)$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: $\quad \bigoplus_ h_i\left(X_\alpha,A_\alpha\right)\to h_i\left(X,A\right)$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 ''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''): $\quad \cdots \to h^i\left(X,A\right) \overset h^i\left(X\right) \overset h^i \left(A\right) \overset h^\left(X,A\right) \to \cdots.$ # Excision: If ''X'' is the union of subcomplexes ''A'' and ''B'', then the inclusion ''f'': (''A'',''A''∩''B'') → (''X'',''B'') induces an isomorphism $\quad h^i\left(X,B\right) \overset h^i\left(A,A\cap B\right)$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: $\quad h^i\left(X,A\right)\to \prod_\alpha h^i\left(X_\alpha,A_\alpha\right)$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. 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 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 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. Some examples of generalized cohomology theories are: * Stable cohomotopy groups $\pi_S^*\left(X\right).$ The corresponding homology theory is used more often: stable homotopy theory, stable homotopy groups $\pi^S_*\left(X\right).$ * Various different flavors of cobordism groups, based on studying a space by considering all maps from it to manifolds: unoriented cobordism $MO^*\left(X\right)$ oriented cobordism $MSO^*\left(X\right),$ complex cobordism $MU^*\left(X\right),$ and so on. Complex cobordism has turned out to be especially powerful in homotopy theory. It is closely related to formal groups, 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: $KO^*\left(X\right)$ (real periodic K-theory), $ko^*\left(X\right)$ (real connective K-theory), $K^*\left(X\right)$ (complex periodic K-theory), $ku^*\left(X\right)$ (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 $E^*\left(X\right)$ 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: *Algebraic K-theory *André–Quillen cohomology *BRST cohomology *Čech cohomology *Coherent sheaf cohomology *Crystalline cohomology *Cyclic cohomology *Deligne cohomology *Equivariant cohomology *Étale cohomology *
Ext groupIn mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic struct ...
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