1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 3. EQUIVARIANT COHOMOLOGY AND THE CARTAN MODEL 313

that i} ( i F) * = e F is multiplication by the equivariant Euler class e F of the normal

bundle to F. further one may show ([30], Section 6, Proposition 8) that the map

I: i:F = Hr(M) I-+ tlJpE;:H*(F) ©Hr

F

is injective. Thus we see that each class T/ E Hr ( M) satisfies

'T}= """' L..,, (. ) 1 Zp *-ZpT} '*

-r ep

FE.r

(3.2)

(by applying i:F to both sides of the equation). Now JM T/ = 7r*T/ (where the

map rr : M --+ pt and rr * : HT( M) --+ Hr is the pushforward in equivariant

cohomology). The result now follows by applying rr* to both sides of (3.2) (since

rr o (ip) = (rrp)* = Jp)·

Proof 3: (Bismut [9]; Witten [41], 2.2.2) Let>. E f!^1 (M) be such that Lx#>. = 0

if and only if X# = 0: for instance we may choose >.(Y) = g(X#, Y) for any tangent

vector Y (where g is any G-invariant metric on M). We observe that if D'T} = 0

then JM 'TJ(X) =JM 'TJ(X)etD>. for any t E IR. Now D>. = d>. - g(X#, X#), so

1 T/(X)etD>. = 1 T/(X)e-tg(X#,x#) L tn(d>.)n /n!.

M M n~O

Taking the limit as t --+ oo we see that the integral reduces to contributions from

points where X# = 0 (i.e. from the components F of the fixed point set of T). A

careful computation yields Theorem 3.10.^2

(^2) The technique used in this proof -introducing a parameter t, showing independence oft by a
cohomological argument and showing localization as t tends to some limit - is by now universal
in geometry and physics. Two of the original examples were Witten's treatment of Morse theory.
in [39] and the heat equation proof of the index theorem [4].