1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

160 D. SALAMON, FLOER HOMOLOGY

and, since df(x) is onto, the dimension of M agrees with the index off.

2.2. The linearized operator

We wish to prove that the moduli spaces M(x-, x+; H, J) are smooth finite di-

mensional manifolds. Hence we must express these spaces as zero sets of functions

between suitable Banach spaces. It is useful to abbreviate the left hand side of (7)

as

• OU OU

OH J(u) =-;;:;--+ J(u)""'il - '\i'Ht(u).

' us ut

This is a vector field along u. Let us fix an element u E M(x-,x+) and consider a

vector space

Xu c C^00 (IR x IR/Z, u*T M)

of all vector fields ~ along u which satisfy a suitable exponential decay condition as

s ., ±oo. Explicitly, think of ~ as a smooth function IR x IR/Z ., TM such that

~(s, t) E Tu(s,t)M· A function near u which also satisfies the limit condition (8) can

be expressed uniquely in the form u' = expu ( ~) for some ~ E Xu. Hence the set of

solutions of (7) and (8) can be expressed as the zero set of a function

Explicitly, :Fu is defined by

:Fu(~)= <I>u(0-^1 BH,J(expu(0)

for ~ E Xu, where <I>u(~) : TuM ___., Texp,,(E,)M denotes parallel transport along

the geodesic T 1-+ expu (TO. The differential of :Fu at 0 is the linear first order

differential operator Du = d:Fu(O) given by

(13)

It turns out that Du is a Fredholm operator between suitable Sobolev completions
of Xu. We introduce the Sobolev norms

(1

00 fl ) l/p

ll~llLP = -oo lo l~lp '

for 1 < p < oo. The corresponding completions of Xu will be denoted by

£P = £P(IR x 31, u*TM), W^1 ·P = W^1 ·P(IR x 51, u*TM).

We shall prove that Du : W^1 ·P ___., LP is a Fredholm operator and express its index
in terms of a suitable Maslov index.

It is useful to simplify the formula for Du by choosing a unitary trivialization

of the vector bundle u*T M ., IR x 31. Such a trivialization takes the form of
a smooth family of vector space isomorphisms (s, t) : IR^2 n ., Tu(s,t)M which
identify the standard symplectic and complex structures w 0 and J 0 on IR^2 n with

(14)