1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

Lecture 3. Equivariant Cohomology and the Cartan Model

Let M be a manifold equipped with the action of a (compact) group G. If the

action of G is free, then M/G is a manifold, but in general it is not. We define a
space Ma (the homotopy quotient) by replacing M by another space with the same
homotopy type on which G acts freely. This is done by introducing a contractible
space EG on which G acts freely and defining

M a = (M x EG)/G.

Ma is well defined up to homotopy equivalence. We may define the equivariant

cohomology^1

H(;(M) = H*(Ma).

Clearly H(;(M) is a module over

H(; ~r H*(BG)

where the classifying space BG is defined as

BG=EG/G.

If G acts freely on M then the natural map from Ma= (M x EG)/ G to M/ G is

a fibration with contractible fibre EG and hence

H(;(M) ~ H*(M/G).

(In fact since we are using cohomology with complex coefficients this is true when

G acts locally freely on M.)

Example 3.1. The action of S^1 on s^2 n-^1 c en (where S^1 acts on each copy of C
by the standard multiplication action) is free. Although s^2 n-^1 is not contractible,
we may form a contractible space

S^00 = {(z 1 , z2, ... ) C C^00 : Zi = 0 for all but finitely many zi, L lzil^2 = 1}

i::'.'.1

on which S^1 acts freely. Thus ES^1 is homotopy equivalent to S^00 and BS^1 is

homotopy equivalent to CP^00 • We have H~(l) = H* (CP^00 ) = C[X], the polynomial

ring in one variable X.

(^1) We shall use cohomology with complex coefficients throughout.
309