1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

210 D. SALAMON, FLOER HOMOLOGY

The tangent space of the ( 6-dimensional) orbit u · G = { u o rp rp E G} at u is given

by

Vertu ={duo() : () E Lie(G)}.

Here we think of() as a vector field on 82 , tangent to a 1-parameter subgroup of

G. The space Vertu is invariant under the obvious action of the isotropy subgroup

Gu by ~ f-) ~ o rp. We shall choose a complement of Vertu which is also invariant

under the action of Gu, by fixing a volume form Wu E 02 (8^2 ) which is invariant

under Gu, and defining

Horu = { ~ E 000 (8^2 , u*TM) : fs

2

(~,duo e)wu =OWE Lie(G)}.

If we extend the notation Wu to the G-orbit of u via Wuoc,o = rp*wu for rp E G then

Horuocp = Horu o rp, Vertuoc,o = Vertu o rp.

Here we have introduced the horizontal spaces as Frechet spaces. In the following

we shall denote the relevant Sobolev completions by Hor~·P. The next Lemma

asserts the existence of multi-valued local slices for the G-action.

Lemma 5.3. Let u : 82 --; M be a smooth function with finite isotropy subgroup

Gu of order m. Then there exists a G-invariant open neighbourhood U = Uk,p c

Wk,p ( 82 , M) of u and a continuous map

p : Uk,p --; 2Hor~·P x G

with the following properties (see Figure 20).

(i) For each v E Uk,p the set p(v) consists of precisely m elements and the

union of the sets p( v) over all v E Uk,p is an open neighbourhood of { 0} x G

in Hor~·P x G.

(ii) If v E Uk,p and ~ E Hor~·P is sufficiently small then

(~,'1/J)Ep(v) {::::::::? v=expu(~)o'ljJ.

(iii) Locally near every v E Uk,p the branches Pi of p are £ times continuously

differentiable as maps from (a neighbourhood av v in) Uk,p to Hor~-e,p x G.

Moreover, the differential

dpi(v): Wk·P(8^2 , v*TM)--; Hor~-l,p x Tw,G

is a compact operator for every v E Uk,p and every i. Here Wi E G is defined

by Pi(v) E Hor~:...l,p x {'I/Ji}.

(iv) If v E Uk,p and (~,'ljJ) E p(v) then

Gv = { 'ljJ-^1 o rp o 'ljJ : rp E Gu, ~ o rp = 0.

Proof. This result. follows from the the implicit function theorem for the map

Hor~·P x G--; Bk,p : (~, '1/J) f-f expu(~) o 'ljJ

which maps {O} x Gu to u. Details can be found in [22]. D

Exercise 5.4. Let p : uk,p --; 2Hor~·PxG be the local slice of Lemma 5.3, and fix
an element v E Uk,p_ If(~, '1/J) E p(v), prove that
p(v)={(~orp-^1 ,rpo'ljJ): rpEGu}· D