1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

s=-oo

LECTURE 2. FREDHOLM THEORY

1 eigenvalues

iof A(s)

I

s

Figure 6. The spectral flow

regular crossings the spectral flow is defined by

μspec(A) = :Z::signf(A,s)

s

s:+oo

where the sum runs over all crossings (cf. Robbin-Salamon [41]).

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Lemma 2.6. Let A(s) be given by {21) and w(s, t) by (15). Then the operator
paths 1-7 A(s) has the same crossings as the symplectic paths 1-7 W(s, 1) and the
crossing forms are isomorphic.

Proof. A function~: IR/Z --t IR^2 n is in the kernel of A(s) if and only if

~(t) = W(s, t)fo, fo E ker (Il - W(s, 1)).

This shows that the crossings are the same. Next we claim that the crossing forms

agree under the natural identification of ker A(s) with ker (Il-W(s, 1) ). This means

that

f(A, s)~

(22)

1

1

(W(s, t)fo, OsS(s, t)W(s, t)fo) dt

(fo, S(s, l)fo)

f(W(·, 1), s)fo

for fo E ker (Il - W(s, 1)), where S(s, t) is defined by

To prove (22) note that

Ot(WT SW) (atwf sw + wT8t(Sw)

-wTSJ 0 Sw - wT Jo8t8sW

-WT S( Os W) - WT Joas (Jo SW)

WT(asS)W.

Integrating over t from 0 to 1 we obtain

W(s, lf S(s, l)W(s, 1) = 1

1

W(s, tf 8sS(s, t)W(s, t) dt

and this implies (22). D