1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

316 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS

The lemma of stationary phase ([21], Section 33) asserts that the oscillatory integral
fmEM eiweiμ(m)X over M has an asymptotic expansion as X---+ oo given by

1

eiweiμ(m)X = L SF(X)(l + 0(1/X)) + O(X-^00 ).

mEM FEF

The first version of the Duistermaat-Heckman theorem (Theorem 4.1) may thus be
reformulated as the assertion that the stationary phase approximation is exact (in
other words the leading order term in the asymptotic expansion gives the answer
exactly for any value of X).

4.2. The natural map "': H(;(M) ---> H*(Mred)

Suppose M is a compact symplectic manifold equipped with a Hamiltonian action
of a compact Lie group G. Suppose 0 is a regular value of the moment map μ.
There is a natural map K: H(;(M)---+ H(Mred) defined by
K,: H(;(M) f-+ H(;(μ-^1 (0)) ~ H(Mred);
it is obviously a ring homomorphism.
Theorem 4.2. The map "' is surjective.
The proof of this theorem ([28], 5.4 and 8.10; see also Section 6 of [30]) uses the
Morse theory of the "Yang-Mills function" lμl^2 : M ---+ ~ to define an equivariant
stratification of M by strata S13 which flow under the gradient flow of -1μ1^2 to a
critical set C 13 of 1μ1^2. One shows that the function 1μ1^2 is equivariantly perfect (i.e.
that the Thom-Gysin (long) exact sequence in equivariant cohomology decomposes
into short exact sequences, so that one may build up the cohomology as

H(;(M) ~ H(;(μ-^1 (0)) EB E9 H(;(S 13 ).

/3#0

Here, the stratification by S 13 has a partial order >; thus one may define an open

dense set U13 = M - U-y>f3S-y which includes the open dense stratum So of points

that flow into μ-^1 (0) (note So retracts onto μ-^1 (0)). The equivariant Thom-Gysin

sequence is

... ---+ H~-^2 d(f3)(S13 ) i!'..; H(;(U 13 )---+ H(;(U 13 - S 13 )---+ · · · ·

To show that the Thom-Gysin sequence splits into short exact sequences, it suffices

to know that the maps ( i13) * are injective. Since i~ ( i13) * is multiplication by the

equivariant Euler class e13 of the normal bundle to S 13 , injectivity follows because

this equivariant Euler class is not a zero divisor (see [28] 5.4 for the proof).

Atiyah and Bott [2] use a similar argument in an infinite dimensional context

to define a stratification of the space of all connections A(:E) on a compact Riemann

surface E, using the Yang-Mills functional JE IFAl^2 (which is equivariant with re-

spect to the action of the gauge group Q): this stratification is used to compute the

Betti numbers of M(n, d).

4.3. Remarks on quantization and representation theory

Definition 4.3. Let M be a symplectic manifold. A prequantum line bundle with

connection on M is a line bundle .C --" M equipped with a connection \7 for which

the curvature F'\l is equal to the symplectic form w.