# Equivariant Cohomology

# Motivation

Let GG be a compact Lie group acting on a topological space XX. If GG acts freely on XX then the quotient space X/GX/G is usually as nice a topological space as XX is itself, i.e. say if XX is a manifold then so it X/GX/G. The motivation for definition of the equivariant cohomology group HG(X)H_{G}^{\ast}(X) is that, should GG act freely on XX, then the equivariant cohomology groups of XX should be just the usual cohomology groups for X/GX/G:

HG(X)=H(X/G),when the action is free. H_{G}^{\ast}(X) = H^{\ast}(X/G),\quad \text{when the action is free.}

Remark

If we let GG act on itself by left multiplication then this implies that

HG(G)=H({pt}). H_{G}^{\ast}(G) = H^{\ast}(\{\text{pt}\}).

When the action of GG is not free however, the space X/GX/G might be somewhat pathological from the point of view of a cohomology theory. The equivariant cohomology groups HG(X)H_{G}^{\ast}(X) are meant to be the most appropriate substitutes for H(X/G)H^{\ast}(X/G).

# Via Classifying Bundles

As cohomology is unchanged by homotopy equivalence, the equivariant cohomology of XX should be the ordinary cohomology of X/GX^{\ast}/G, where XX^{\ast} is a topological space homotopy equivalent to XX, and on which GG acts freely. To construct such a space, we can take the product X=X×EX^{\ast} = X \times E where EE is a contractible space on which GG acts freely. Then the standard way of defining the equivariant cohomology groups of XX is by:

Definition

The equivariant cohomology groups of a manifold XX with respect to a Lie group GG is

HG(X):=H((X×E)/G), H_{G}^{\ast}(X) := H^{\ast}\big( (X \times E) / G \big),

where EE is any contractible space on which GG acts freely.

# Independence of EE

To show that the definition does not depend on the choice of EE, we first note that if GG acts freely on XX then the projection

pX:X×EX p_{X} : X \times E \rightarrow X

onto the first factor gives rise to a map

(X×E)/GX/G (X \times E)/G \rightarrow X/G

which is a fibration with typical fibre EE. As EE is contractible, we have that

HG(X)=H((X×E)/G)=H(X/G), H_{G}^{\ast}(X) = H^{\ast}\big( (X \times E) / G \big) = H^{\ast}(X/G),

which is a property that we require. Also since the above contruction gives rise to a fibre bundle over X/GX/G with contractible fibre, it admits a global cross-section

s:X/G(X×E)/G. s: X/G \rightarrow (X \times E)/G.

The projection

pE:X×EE p_{E} : X \times E \rightarrow E

onto the second factor gives rise to a map

(X×E)/GE/G. (X \times E) / G \rightarrow E/G.

The composition of this projection along with the section ss gives rise to a map

f:=sp:X/GE/G. f := s \circ p : X/G \rightarrow E / G.

Let

qX:XX/G,qE:EE/G, q_{X} : X \rightarrow X / G,\qquad q_{E} : E \rightarrow E / G,

be the projections of XX and EE onto their quotient spaces under the respective GG-action.

Proposition

Suppose that GG acts freely on XX and that EE is a contractible space on which GG acts freely. Any cross-section s:X/G(X×E)/Gs : X / G \rightarrow (X \times E) / G determines a unique GG-equivariant map

h:XE h : X \rightarrow E

which makes the diagram
(insert diagram)
commute.

Convsersely, any GG-equivariant map h:XEh : X \rightarrow E determines a section s:X/G(X×E)/Gs : X/G \rightarrow (X \times E) / G and a map ff which makes the above diagram commute.

Any two such sections are homotopic and hence the homotopy class of (f,h)(f,h) is unique, independent of the choice of ss.

This proposition is usually stated as a theorem about principal bundles: since GG acts freely on XXwe can consider XX as a principal bundle over

Y:=X/G. Y := X/G.

Similarly we can regard EE as a principal bundle over

B:=E/G. B := E/G.

Then the above proposuition is equivalent to the following "classification theorem" for principal bundles:

Theorem

Let YY be a topological space and π:XY\pi : X \rightarrow Y a principal GG-bundle. THen there exists a map

f:YB f : Y \rightarrow B

and an isomorphism of principal bundles

Φ:XfE \Phi : X \rightarrow f^{\ast}E

where fEf^{\ast}E is the pull-back of the bundle EBE \rightarrow B to YY. Moreover ff and Φ\Phi 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 GG-bundles and homotopy classes of mappings f:YBf : Y \rightarrow B. In other words, the theorem reduces the classification problem for principal GG-bundles over YY to the homotopy problem of classifying maps of YY into BB up to homotopy. For this reason the space BB is called the classifying space for GG and the bundle EBE \rightarrow B is called the classifying bundle.

An important consequence of the above theorem is the following:

Theorem

If E1E_{1} and E2E_{2} are contractible spaces on which GG acts freely, they are equivalent as GG-spaces. I.e., there exist GG-equivariant maps

ϕ1:E1E2,ψ:E2E1, \phi_{1}: E_{1} \rightarrow E_{2},\qquad \psi : E_{2} \rightarrow E_{1},

with GG-equivariant homotopies

ψϕidE1,ϕψidE2. \psi \circ \phi \simeq \id_{E_{1}},\qquad \phi \circ \psi \simeq \id_{E_{2}}.

Corollary

The definition

HG(X):=H((X×E)/G) H_{G}^{\ast}(X) := H^{\ast}\big( (X \times E) / G \big)

is independent of the choice of EE.

# Existence of Classifying Spaces

The previous section asserts that our definition of equivariant cohomology does not depend on which EE that we choose. However, it is still needed to be shown that such an EE exists, i.e given a compact Lie group GG can we find a contractible space EE on which GG acts freely? If GG is a subgroup of a compact Lie group KK and also if we have found an EE that works for KK, then restricting the KK-action to the subgroup GG produces a free GG-action. Every compact Lie group has a faithful linear representation, meaning that it can be embedded as a subgroup of U(n)U(n) for large enough nn. Thus it suffices to construct a space EE which is contractible and on which U(n)U(n) acts freely.

Let VV be a infinite-dimensional separable Hilbert space, say take

V=L2[0,), V = L^{2}[0,\infty),

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 EE consist of the set of all nn-tuples

v=(v1,,vn),viV,(vi,vj)=δij. \boldsymbol{v} = (v_{1},\ldots, v_{n}),\quad v_{i} \in V,\quad (v_{i},v_{j}) = \delta_{ij}.

The group U(n)U(n) acts on EE by

Av=w=(w1,,wn),wi=jaijvj, A\boldsymbol{v} = \boldsymbol{w} = (w_{1},\ldots, w_{n}),\quad w_{i} = \sum_{j} a_{ij}v_{j},

and this action is clearly free.

Proposition

The space EE is contractible.

Thus, with this proposition in hand, for any compact Lie group there exists a classifying space EE on which GG acts freely.