1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

328 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS

Exercise 2 (Lectures 3-5)

l. (Equivariant cohomology of subgroups) (a) Let G be a compact Lie

group. Suppose H C G is a Lie subgroup of G. Show that there is a map

H(:;(M) __, H'H(M).

(Either use the Cartan model, or else use the existence of a fibration M x H

EH__, M xc EG. This fibration exists since, as EG is a contractible space

on which G acts freely, it is also a contractible space on which H acts freely,

and hence may be identified with EH.)

(b) Show that in particular there is a map H(;(M) __, HT(M) where T is

the maximal torus of G.

( c) In (b), let M be a point. Using the Cartan model to identify H(:; (pt) ~

5(g*)c and HT(pt) ~ 5(t*), show that the image of the map H(;(pt) __,

HT(pt) (which is in fact isomorphic to H(;(pt)) is 5(t*)w C 5(t*).

(d) Show that if G = 5U(2) and T = U(l), the image of H(;(pt) in HT(pt) ~

JR[X] is the polynomials of even degree in X.

2. (Functorial properties of equivariant cohomology) Show that if M 1

and M 2 are two manifolds equipped with action~ of G and f : M 1 __, M 2 is

a G-equivariant map then it induces a map f* : H(:;(M2) __, H(;(M 1 ). (This

correspondence has the property that if M3 is another manifold with a G

action and g : M2 __, M3 is G equivariant then (g o f) = f o g*.)

3. (a) Show that if G is a compact Lie group and H is a Lie subgroup of G
   then
   H(:;(G/ H) ~ H'H(pt).
   (Hint: Rewrite EG x c G / H as a quotient by H.)
   (b) Let G = 50(3) and H = U(l). What is the ring H(;(5^2 ) where 50(3)

acts on 52 by rotation? (Hint: write 52 = 50(3)/U(l). )

4. (Functorial properties of pushforward) Let M be a G-manifold. Show
   that if a E 5(g)G and 7r a is its inclusion as an element of D.(;(M), and
   {3 E D.(;(M), then

JM ({37r*a) =(JM {3)a.

It follows that the map JM : H(;(M) __, HG(pt) is a homomorphism of

H(;(pt)-modules.

5. (Duistermaat-Heckman for the two-sphere)
   Let T = U(l) act on M = 52 by rotation about the z axis, so that the

moment map is μx(¢, z) = Xz.

(a) Compute the integral

by elementary methods.

(b) Compute the same integral using the Duistermaat-Heckman theorem.