# Equivariant Cohomology
Let be a compact Lie group acting on a topological space . If acts freely on then the quotient space is usually as nice a topological space as is itself, i.e. say if is a manifold then so it . The motivation for definition of the equivariant cohomology group is that, should act freely on , then the equivariant cohomology groups of should be just the usual cohomology groups for :
If we let act on itself by left multiplication then this implies that
When the action of is not free however, the space might be somewhat pathological from the point of view of a cohomology theory. The equivariant cohomology groups are meant to be the most appropriate substitutes for .
# Via Classifying Bundles
As cohomology is unchanged by homotopy equivalence, the equivariant cohomology of should be the ordinary cohomology of , where is a topological space homotopy equivalent to , and on which acts freely. To construct such a space, we can take the product where is a contractible space on which acts freely. Then the standard way of defining the equivariant cohomology groups of is by:
The equivariant cohomology groups of a manifold with respect to a Lie group is
where is any contractible space on which acts freely.
# Independence of
To show that the definition does not depend on the choice of , we first note that if acts freely on then the projection
onto the first factor gives rise to a map
which is a fibration with typical fibre . As is contractible, we have that
which is a property that we require. Also since the above contruction gives rise to a fibre bundle over with contractible fibre, it admits a global cross-section
onto the second factor gives rise to a map
The composition of this projection along with the section gives rise to a map
be the projections of and onto their quotient spaces under the respective -action.
Suppose that acts freely on and that is a contractible space on which acts freely. Any cross-section determines a unique -equivariant map
which makes the diagram
Convsersely, any -equivariant map determines a section and a map which makes the above diagram commute.
Any two such sections are homotopic and hence the homotopy class of is unique, independent of the choice of .
This proposition is usually stated as a theorem about principal bundles: since acts freely on we can consider as a principal bundle over
Similarly we can regard as a principal bundle over
Then the above proposuition is equivalent to the following "classification theorem" for principal bundles:
Let be a topological space and a principal -bundle. THen there exists a map
and an isomorphism of principal bundles
where is the pull-back of the bundle to . Moreover and are unique up to homotopy.
This theorem can be reformulated as saying that there is a one-to-one correspondence between equivalence classes of principal -bundles and homotopy classes of mappings . In other words, the theorem reduces the classification problem for principal -bundles over to the homotopy problem of classifying maps of into up to homotopy. For this reason the space is called the classifying space for and the bundle is called the classifying bundle.
An important consequence of the above theorem is the following:
If and are contractible spaces on which acts freely, they are equivalent as -spaces. I.e., there exist -equivariant maps
with -equivariant homotopies
is independent of the choice of .
# Existence of Classifying Spaces
The previous section asserts that our definition of equivariant cohomology does not depend on which that we choose. However, it is still needed to be shown that such an exists, i.e given a compact Lie group can we find a contractible space on which acts freely? If is a subgroup of a compact Lie group and also if we have found an that works for , then restricting the -action to the subgroup produces a free -action. Every compact Lie group has a faithful linear representation, meaning that it can be embedded as a subgroup of for large enough . Thus it suffices to construct a space which is contractible and on which acts freely.
Let be a infinite-dimensional separable Hilbert space, say take
the space of square-integrable functions on the positive real number line, relative to the Lebesgue measure (any separable Hilbert space is isomorphic to one another).
Let consist of the set of all -tuples
The group acts on by
and this action is clearly free.
The space is contractible.
Thus, with this proposition in hand, for any compact Lie group there exists a classifying space on which acts freely.