In
mathematics, equivariant cohomology (or ''Borel cohomology'') is a cohomology theory from
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 invariants that classify topological spaces up to homeomorphism, though usually most classify ...
which applies to
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s with a ''
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
''. It can be viewed as a common generalization of
group cohomology and an ordinary
cohomology theory
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
. Specifically, the equivariant cohomology ring of a space
with action of a topological group
is defined as the ordinary
cohomology ring with coefficient ring
of the
homotopy quotient :
:
If
is the
trivial group
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...
, this is the ordinary
cohomology ring of
, whereas if
is
contractible
In mathematics, a topological space ''X'' is contractible if the 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 a point within th ...
, it reduces to the cohomology ring of the
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
(that is, the group cohomology of
when ''G'' is finite.) If ''G'' acts freely on ''X'', then the canonical map
is a homotopy equivalence and so one gets:
Definitions
It is also possible to define the equivariant cohomology
of
with coefficients in a
-module ''A''; these are
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s.
This construction is the analogue of cohomology with local coefficients.
If ''X'' is a manifold, ''G'' a compact Lie group and
is the field of real numbers or the field of complex numbers (the most typical situation), then the above cohomology may be computed using the so-called Cartan model (see
equivariant differential forms.)
The construction should not be confused with other cohomology theories,
such as
Bredon cohomology or the cohomology of
invariant differential forms: if ''G'' is a compact Lie group, then, by the averaging argument, any form may be made invariant; thus, cohomology of invariant differential forms does not yield new information.
Koszul duality In mathematics, Koszul duality, named after the French mathematician Jean-Louis Koszul, is any of various kinds of dualities found in representation theory of Lie algebras, abstract algebras (semisimple algebra) and topology (e.g., equivariant cohom ...
is known to hold between equivariant cohomology and ordinary cohomology.
Relation with groupoid cohomology
For a
Lie groupoid In mathematics, a Lie groupoid is a groupoid where the set \operatorname of objects and the set \operatorname of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smo ...