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This is to certify that the

dissertation entitled

The spectral flow, the Maslov index, and
decompositions of manifolds

presented by

Liviu I. Nicolaescu

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Mathematics

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THE SPECTRAL FLOW,THE MASLOV
INDEX AND DECOMPOSITIONS OF
MANIFOLDS

By

Liviu I. Nicolaescu

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the Degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1994

Thomas Parker, Major Professor

ABSTRACT

THE SPECTRAL FLOW,THE MASLOV INDEX AND DECOMPOSITIONS
OF MANIFOLDS

BY

LIVIU I. NICOLAESCU

Let E —+ M be a Clifford bundle over a compact oriented Riemann manifold
(M,g) and let D(t) : C°°(£) —’ C°°(£) be a path of selfadjoint Dirac operators.
M is divided into two manifolds-with-boundary by a hypersurface E C M. Set
80 = 8'2: Dj(t) = D(t)|M’ Aj(t) = kerDJ-(t)|2j=1,2. The Clifford multiplication
defines a symplectic structure in L2(80) such that Aj(t) are (infinite dimensional)
lagrangian subspaces. The main result of this thesis shows that the spectral flow
of the family D(t) is equal to the Maslov index of the (continuous) path t H
(A1(t),A2(t)). We then show that an adiabatic deformation of the neck reduces

the computation of the Maslov index to a finite dimensional situation.

To my parents and my wife

ACKNOWLEDGEMENT

I am deeply indebted to my advisor, Professor Thomas Parker. The conver-
sations with him have dramatically changed the way I view mathematics. His
patience, insights and advises were and will always be an excellent guide and sup-

port.

It makes me a great pleasure to thank Professor Ronald Fintushel. The discus-
sions with him taught me not only gauge theory but they always transmitted me
part of his enthusiasm and passion for mathematics. I spent many exciting hours
with Professor Jon Wolfson discussing about symplectic geometry and mathemat-
ics in general. I want to thank him for his advises and encouragements which meant
a lot for an aspiring grad student. While working on the results in this thesis I
benefitted from the correspondence with Kryzstof Wojciechovski who supplied me
with valuable materials and whose interest in my research kept me going. I want

to thank both him and Paul Kirk for the stimulating days I spent in Indiana.

Chairman Richard Phillips always has had an open door for graduate students
in general and for me in particular. I want to thank him for his support during
my studies at MSU. This great MSU experience would not have happend if it were
not for Professor Charles Seebeck. He took a chance with me by bringing me here
and I am very much indebted to him. His support and advise during my first year

here meant very much to me.

I owe very much to my wife who stood by me at all times. Her warmth helped
me go through many rainy days. At last but not the least I want to thank my
parents for their love and support they invested in me which shaped the person I

am DOW.

iv

Contents

1 Introduction

2 Infinite dimensional lagrangian subspaces

3 The Maslov index in infinite dimensions

4 Boundary value problems for selfadjoint Dirac operators
5 The Maslov index and the spectral flow

6 Adiabatic limits of CD spaces

7 The Conley-Zehnder index

A The proof of Proposition 5.6

References

14

25

37

48

67

78

85

List of Figures

2.1

4.1

4.2

5.1

6.1

6.2

6.3

6.4

6.5

7.1

7.2

7.3

Lagrangian subspaces can be viewed as graphs of symmetric operators 9

The metric is cylindrical in a neighborhood of the boundary . . . . 26
A metrically nice splitting ....................... 35

The cutoff function a ......................... 45
Adiabatic deformation of the neck ................... 49
Hyperbolic flow ............................. 52
Dynamics on A(l) ............................ 52
AT is the graph of B, 2 .15, + h, .................. 59
A Morse flow on the lagrangian grassmanian ............. 65
Splitting Sl ............................... 71
Intersecting the resonance divisor ................... 72
An easier way to compute the Conley-Zehnder index ........ 76

vi

Chapter 1

Introduction

Consider a closed, compact oriented Riemann manifold (M,g) and a Clifford bundle
E —» M over M. The spectral flow of a smooth path of selfadjoint Dirac operators
D‘ : C°°(E) —§ C°°(E) is the integer obtained by counting, with sign, the number
of eigenvalues of D‘ that cross zero as t varies; it is a homotopy invariant of the
path (cf. [A8]). The aim of this thesis is to describe the spectral flow in terms of

a decomposition of the manifold.

More precisely, suppose that M is divided into two manifolds-with-boundary
M1 and M2 by an oriented hypersurface E C M. Assume that in a tubular
neighborhood N of 2 the metric is a product and the operators D‘ have the

“cylindrical” form

0* = c(ds)(a/08 + 03) (1.1)

where s is the normal coordinate in N, C(ds) is the Clifford multiplication by ds,
and D3 is independent of 8. Set 80 = 8|E and denote by Di and D; the restriction

Of 0‘ to M1 and M2.

The kernels of D;- are infinite dimensional spaces of solutions of Dj-ib = 0 on

1

Mj. Restriction to 2 gives the “Cauchy data spaces”(CD spaces)
A1(t) = KerDiIE, A2(t) = KerDéI,3

in L2(Eo). Note that the intersection A1(t) fl A2(t) is the finite dimensional space

of solutions of DW = 0 on M.

This setup has a rich symplectic structure. Multiplication by C(dS) introduces
a complex structure in L2(£o) and hence a symplectic structure in this space. The
CD spaces Aj(t) are then infinite dimensional lagrangian subspaces of L2(£o) that
vary smoothly with t, and the pair (A1(t),A2(t)) is a Fredholm pair (as defined
in chapter 2). As in the finite dimensional case, one can associate to a path of
Fredholm pairs of lagrangians an integer called the Maslov index. The main result
of this thesis (Theorem 5.14) states that this Maslov index equals the spectral flow

of the family D‘.

The lagrangians defined by the CD spaces are infinite dimensional, but the
setup can be reduced to finite dimensional symplectic geometry by “stretching the
neck”. This is done by changing the metric on M to one in which the neck is
isometric to a long cylinder (—r, r) x 2. We study the Cauchy data spaces in the
adiabatic limits 1' —» 00. These limits exist if we assume D is “ neck-compatible”,
i.e. is cylindrical and the operator Do in ( 1.1) is selfadjoint. If, moreover, certain
nondegeneracy conditions are satisfied these limits have a nice description and our
Cauchy data spaces A ,(t) stabilize to asymptotic Cauchy data spaces A? . These
limiting spaces arise naturally in the Atiyah-Patodi-Singer index problem([APSl-
3]). A related adiabatic analysis was considered in [CLM2]. After performing
this adiabatic deformation we can reduce the Maslov index computation to a finite

dimensional situation by passing to a symplectic quotient. This generalizes a recent

result of Yoshida [Y] in the context of Floer’s instanton homology.

The thesis consists of seven chapters. In chapter 2 we translate some basic facts
of finite-dimensional symplectic topology into infinite dimensions. The main result
here is the homotopical description of the space of Fredholm pairs of lagrangians:

it has the homotopy type of the classifying space of K0”.

Chapter 3 deals with the Maslov index in infinite dimensions. Using Arnold’s
definition ([Ar]) as a model, we define it as an intersection number. We then derive

some computational formulae which play a crucial part later.

Chapter 4 contains the main analytical technicalities of this paper. Many of
these results are known but we have reformulated them in a symplectic context

(see [BW4] for an extended presentation of this subject).

Chapter 5 contains contains our main result: The Maslov index equals the
spectral flow. The idea of the proof is to reduce the general problem via successive
homotopies to a simple situation. For this we rely on a genericity result first used
by Floer ([F]) in the context of symplectic homology (we give a complete proof
in an Appendix). After reducing to the case of piecewise affine homotopies, the
theorem follows by an integration by parts formula. Again, this has an elegant

symplectic interpretation.

In chapter 6 we take up the problem of stretching the neck. This entails study-
ing the behavior of the Cauchy data spaces of a neck-compatible Dirac on a mani-
fold M as the length of the neck tends to infinity. We begin by studying a related
finite dimensional problem. Namely, suppose that A is a 2n x 2n symmetric matrix
that anticommutes with the canonical complex structure J on on R“. We then

get a l-parameter group of symplectic transformations r—ie'“, and hence a flow

on the lagrangian grassmanian A(n) of R2". In Corollary 6.1 we show that each
trajectory in A(n) has a unique limit point as r —» 00; this limit is an A-invariant
lagrangian in R2". This follows from a simple trick we learned from Tom Parker.
We then return to the infinite dimensional problem, where we can regard the CD
spaces as infinite dimensional lagrangians evolving by the “flow” r—re"D° as the
neck length r —> 00. By passing to a carefully defined symplectic quotient, we re-
late this to the above finite dimensional situation. This yields Theorem 6.9, which
shows that as the neck length r —i 00 the Cauchy data spaces stabilize to limiting
infinite dimensional lagrangians. We can then obtain the Maslov index from a

computation in the finite dimensional symplectic quotient (Corollary 6.14).

Finally in chapter 7 we present a simple application where the splitting formula
of Theorem 5.14 is used to identify the Conley-Zehnder index of [OZ] with a spectral

flow.

We would like to mention that Tom Mrowka informed the author that he re-
cently proved these results using a similar approach. After this work was completed
the author learned that Ulrich Bunke independently obtained a splitting formula
for the spectral flow (see [Bu2]) as consequence of a gluing result for the eta function

of a neck-compatible Dirac (see [Bu1]). The results of this thesis were announced

in [N1].

Chapter 2

Infinite dimensional lagrangian
subspaces

In this chapter we study lagrangian subspaces in an infinite dimensional symplectic
space. In contrast to the finite dimensional situation, the grassmanian of lagrangian
subspaces is contractible. A related, but more topologically interesting space is the
space of Fredholm pairs of lagrangians. We will show this is a classifying space for
K01 .

Let H be a separable real Hilbert space with inner product ( , ). We will denote
the *-algebra of bounded linear operators on H by B(H). Let GL(H) be the group
of invertible elements in B(H) and O(H) be the subgroup of bounded orthogonal
operators. For A, B E B(H) define the commutator and the anticommutator as

usual

[A,B] = AB — BA, {A, B} = AB + BA.

Fix once and for all a complex structure on H, that is, an operator J E O(H) with

J 2 = —1. Thus H becomes a symplectic space with symplectic form
w(way) = (Jxay) any E H-

We can then introduce the basic notions of symplectic geometry. Let W be a sub-

5

6

space of H (the word subspace will always mean closed subspace). Its annihilator

is the subspace

W°={y€H; w(w,y)=0 VwEW}.

It is easily seen that W0 = J Wl where WJ- is the orthogonal complement of W
in H.

Definition A subspace W of H is called isotropic if W C W0, coisotropic if
W0 C W, and lagrangian if W0 = W. Equivalently, W is lagrangian if W-‘- =
J W.

Let .C = £1 be the set of lagrangian subspaces of H. To topologize [I we
identify it with a space of operators using the following construction. Associated
to each lagrangian are three operators: the orthogonal projection PL onto L, the
complementary projection Q L = I — PL onto the orthogonal complement of L and

the conjugation operator (reflection through L)
CL=PL—QL=2PL—I.
Note that C = CL satisfies
C = C‘, 02 = I, {C,J} = O. (2.1)

It is easy to see that if C satisfies (2.1) then Ker(I - C) is a lagrangian subspace

with projection PL = 1/2(I + C). Thus we can identify £1 with

C; = {C ;C satisfies (2.1)} (2.2)

and topologize it using the operator norm. We will use this identification £1 = C j

frequently below.

Now thinking of (H,J) as a complex Hilbert space, the unitary group
UJ(H) = {U6 O(H); [UaJl=0}

is a topological group which is contractible by Kuiper’s theorem ([Ku]) and which

acts on on .6 by

C H UCU'I.

This action is transitive (just as in the finite dimensional case, cf. [GS]). The

stabilizer of C is

06: {Hero ;[U,C]=0}.

By standard arguments ([BW2] or [A8]) we have a fibration
O(L) —> 11.; —> .C

where, again by Kuiper’s theorem, O(L) is contractible. The long exact sequence

in homotopy implies the following result.

Proposition 2.1 .C is contractible.

Thus in infinite dimensions .C has no interesting topology. To get something

interesting we will consider
5(2) = {(A1,A2) 6 £2 ; (A1,A2) Fredholm pair}.

Recall that a pair of (V,W) of infinite dimensional subspaces of H is called Fredholm
if both subspaces have infinite codimension, V+W is closed and both dim( V D W)

and codim (V+W) are finite. The Fredholm index of this pair is defined as

aw, W) = dim(VnW) — dim(V + W).

8

(For basic facts about F redholm pairs we refer to [C] or [K] ). Note that F redholm

pairs of lagrangians automatically have index 0 since

i(A1, A2) = dim(A1nAa)—dim(A,i, Ag) = dim(AlnA2)—dimJ(AlnA2) = o. (2.3)

We can also describe U2) in terms of conjugation operators. By Lemma 2.6 of

[BW2], (A1, A2) 6 11(2) iff the corresponding conjugations satisfy
01 + C2 6 IC
where [C is the space of compact operators on H. Thus
U2) = {(01,02) 6 C2 ;C1 + C2 6 1C}.
Now fix Co E C. We have a fibration
£0 ¢—» [1(2) 3* L:

where p(C1,C2) = 01 and £0 = p“(Co) = (—C0 + IC) D C. Since L is contractible

we get a weak homotopy equivalence
£0) 2’ £0.

Setux=u10(I+IC) andforCECset Og,c=(I+IC)00c.

Theorem 2.2 There exists a weak homotopy equivalence.
£0 ’5 U(oo)/0(oo)

where

U(oo) = 1131mm 0(00) = 1111100.).

 

 

 

Ao

 

 

Figure 2.1: Lagrangian subspaces can be viewed as graphs of symmetric operators

Proof: The proof will be carried out in several steps, with some intervening

Lemmas.
Step 1 .Co is path connected.

Associated to each finite dimensional subspace V C A0 is a subspace
£0(V) = {A 6 £0 ;AflAo C V}

of £0; these define a filtration of £0. To show that £0 is connected it suffices
to show that each £0(V) is connected. Now in finite dimensibns the space of
lagrangians in V+JV is connected (see [GS]). Hence any lagrangian in £0(V) can
be connected in £o(V) to a lagrangian in £o(0). Thus it suffices to show that
£o(0) is connected. This follows immediately from the next lemma, which gives an
alternate description of £o(0). The idea, which is standard in the finite dimensional

case, is to regard lagrangian subspaces as the graphs of symmetric operators (cf.

[Ar], [GS]).

Lemma 2.3 There is an identification

[30(0) 9-: {selfadjoint operators J A0 —> J A0}

10

and hence £o(0) is contractible.

Proof: Suppose that (A,Ao) is a Fredholm pair of transversal lagrangians. Let
P = PA, and Q = I-P. We deduce that H = A + A0. In particular this implies that
Q(A) = JAo (see Fig. 2.1). Using the fact that AflAo = 0 we see that Q : A —+ JAo
is also injective. The open mapping theorem implies that Q is an isomorphism.

Construct the operator A : J A0 —> J A0 by
li H Q'll‘L H PQ’IIi H JPQ‘lll.

Clearly A is a bounded operator (by the closed graph theorem). Note that:
(i) each u E A can be uniquely written as u = Ii — JAIJ- where 1* = Qu E JAG;
(ii) the condition that L is lagrangian is equivalent to A being selfadjoint.

Conversely, given a selfadjoint operator A : J A0 —» J A0 its “graph”
AA = {Ii — JAIL; 1" E JAG}
is a lagrangian. Note that A A 0 A0 = 0. Now consider the operator
A: H —+ H , A(IJ'L) = JAIJ’ , (All) 6 A0 69 JAo.

One sees that Range(I — A) = A0 + A A so that the transversality of the pair
(A A, A0) is equivalent to the surjectivity of I — A. Since AZ = 0 for any selfadjoint
A : J A0 —> J A0 we deduce that for any such A, I —A is invertible. Hence (A A, A0)

is a transversal F redholm pair. 0
Step 2: If C1, C2 6 £0 satisfy ||C1 — C2” < 2 then there is a T in

or,C = {T e GL(H) n (I +IC); [1: J] = 0}

11

such that

cazrarq. (an

Following [W] we set T = I + 1/2(Cl - C2)C1. Then T is invertible since ”(C1 —
C2)C1]| < 2, and T commutes with J because C1 and 02 anticormnute with J. On
the other hand, 01 and C2 lie in —C + K, so T 6 GL;. A simple computation
shows that (2.4) holds.

Step 3: For each pair C1, C2 E £0 there is a T 6 CL): such that
Cg = TClT'd. (2.5)

This follows from Step 2 and the path-connectedness of £0; the details are left to

the reader.

To proceed further we need the following technical result.
Lemma 2.4 IfT 6 CL): then (T'T)1/2 E GLK.

Proof: Set 5 == T‘T. Clearly 31/2 6 GL(H) and 31/2 commutes with J. We
have to show that 51/2 E I + K. Then S E I + K. To find 31/2 we use Newton’s

iteration as in [Ku]
50 = I , Sn“ =1/2(S,, + 5:15).

Note that this iteration is well defined since all S,’,s are invertible (they are positive
selfadjoint operators with their spectra bounded away from 0). One sees induc-

tively that the right hand side of the iteration is an affine combination of terms in

I+K. Thus Suzzlim 5,. E I+K. 0
Step 4 For each pair Cl, C2 6 .Co there is a U E U; such that

Q=Uaut on

12

We follow the idea in [B], Prop.4.6.5. Consider T E GL5 as in (2.5). Then
TC; = C2 T and Cl T" = T‘ 02. It follows that Cl commutes with S = S = T‘T,
and hence with 31/2. Setting U = T(TT")’1/2, we clearly have U‘U = I, and

U E GL5 by Lemma 2.4. Therefore U is in U5 and satisfies (2.6).

Step 4 shows that L15 acts transitively on .C. For C 6 £0 the stabilizer of this

action is 050. Thus

£0 EU5/0pc , C E Co. (2.7)

Step 5 There are homotopy equivalences
UK; 2’ GL5 2’ GL(OO, C) , 05,0 ’5 GL(OO,R). (2.8)

The proof of 115 E GL5 is identical to the proof of Lemma 2.9 of [BW2]. It
essentially uses the polar decomposition which by Lemma 2.4 is an internal de-
composition in GL5, followed by an affine deformation of the positive symmetric
term of the polarization. I + K is an affine space so this deformation stays within

GL5. Then by the results of Palais [P1] we have a homotopy equivalence
GL5 = GL(oo,C).
The second part is completely analogous. Classically
U(oo) E GL(oo,C) , 0(00) 9-: GL(oo,R) homotopically. (2.9)

Theorem 2.2 follows form (2.7), (2.8) and (2.9). 0

Remark 2.5 A related result was proved in [W], [B W3]. In that context K repre-

sents compact pseudodifferential operators in some complex L2 space.

13

The above arguments apply in finite dimensions to show that the grassmanian
A(n) of lagrangians in C" is diffeomorphic to U(n)/O(n). Taking the direct limits

over the embeddings A(n) H A(n + 1) then gives
A(oo) = l_i_mA(n) E U(oo)/0(oo).

Hence we get

Corollary 2.6

U2) g £0 “=’ U(oo)/0(oo) E A(oo).

It is known that U / O is a classifying space for KO1 (cf. [Kar]). On the other hand
Atiyah-Singer [AS] have shown that this classifying space can also be identified (up
to homotopy) with the space of selfadjoint Fredholm operators on a real Hilbert
space. Its fundamental group is isomorphic to Z. The isomorphism is given by the

the spectral flow (of a loop of selfadjoint F redholm operators). Obviously

and the isomorphism is given by the Maslov index. Thus Corollary 2.6 displays the
double nature of U2): the operator theoretic nature and the symplectic nature. In

the sequel we will further analyze this duality.

Chapter 3

The Maslov index in infinite
dimensions

The purpose of this chapter is to provide a more computational description of
the Maslov index introduced in the previous chapter. In the finite dimensional
situation there are many excellent presentations of the Maslov index (see e.g.[Ar],
[CLMl], [DI-2], [GS], [RS]). However, all these assume the finite dimensionality,
especially when dealing with orientability questions. For a Banach manifold ori-
entability is a delicate question. To avoid this issue we will give a meaning to
a local intersection number without any elaborate considerations of orientability.

Our approach is inspired from Arnold’s description of the finite dimensional index
([Ar])-
Let (H,J) be a Hilbert space with a complex structure and consider a lagrangian

A0 = Ker(I — Co) specified by the conjugation Co. The next several lemmas

describe the geometry of the space

£0 = {A 6 £ / (A0, A) is a Fredholm pair}.

Lemma 3.1 £0 is a smooth Banach manifold modelled on the space Sym(JAo) of

bounded symmetric operators on J A0.

14

15

Proof To each finite dimensional subspace V of A0 we associate an orthogonal
operator Iv commuting with J by

I(v)- Jv forvEV
V — v forvEAflV‘L.

and the open subset

Dv={AE£ AnlvAo=0}.

Thus Dy = IVLB, where
£3: {AEfio/A0A0=O}.

is the dense open set of transverse pairs (in particular, D0 = C5). Notice that
Iv E 115, so (IVA,A0) is a Fredholm pair and thus Dv C CO. The sets Dy cover
£0: ifAELo thenAE’Dv for V = AflAo.

The isomorphism of Lemma 2.3 is a map \Ilo : D0 = L; -—> Sym(JAo). For
other V, set

\Ilv = We 0 1‘71 :Dv —> Sym(JAo).
Then the collection

{(DV,\IIV) ; V e V, w 20v —> Sym(JAo)}

forms an atlas of £0. The verification that the transition functions are smooth is
accomplished by writing the conjugation operator C associated to a lagrangian in

terms of of these coordinates. Thus if A E [.3 and we identify A0 with J A0 via J

_ 0 ‘le
J-lal 0]

then the conjugation associated to A, C : A0 EB A0 —> A0 EB A0 can be described as

so that J becomes

0 _ [ (1+ 52)-1(1— 32) 2(1+52)-15 [
" 2(1+ s2)-ls —(1 + 52)-1(1— 52)

16

where S = \II0(A). The details are left to the reader. 0

The manifold Co is filtered by the subspaces
3‘ = {A 6 £0; dim(Afl A0) = m}.

In fact, these are subvarieties, as follows. Note that £3" is covered by charts of
the form Dv with dimV = m. Fix one such chart and write 5' = \Ilv. By an
elementary argument of Arnold ([Ar] Lemma 3.3.3) one sees that A 6 DV lies in
£5” if and only if

(SJu,.Iv) = 0 for all 11,1) 6 V. (3.1)

Since 5 is symmetric and dim V = m this describes £5" in this chart as the solution

set of m(m + l) / 2 algebraic equations.

In particular, if A 6 £3, then AflAo is a l-dimensional space V0 2 Re, A 6 Dvo,
and S = \IIVO(A) then
(SJe, Je) = 0. (3.2)

Corollary 3.2 The closure Z}: is a codimension 1 subvariety of £0 called the res-

onance divisor. It is stratified by subvarieties £5” of codimension m(m+1)/2.

We may think of L—(l, as a divisor in £0 defining an element in H1(£o, Z) E Z
dual to the generator of H1(£o, Z). Dually, given a loop 7 in £0, we may think of
its Maslov index p(7) as being the intersection number 7 n £0. Most of the rest of
this chapter is devoted to making this intuition rigorous. We will first show that if
a path 7 intersects £5 transversally, one can associated a sign to each intersection
point. The sum of then intersection numbers is a homotopy invariant of the path.
As a byproduct, we will get several formulas for the local intersection number;

these will be extremely useful later.

17

Consider the vector field x over £0 defined by

X(A) = 5’, (eJ‘A).

Proposition 3.3 X defines a transversal orientation on C3.

Proof This follows easily from a computation of Arnold ([Ar] Lemma 3.5.3). Con-
sider A 6 £0 and assume A lies in a coordinate chart Dv, dim V = 1. If S = \IIV(A)

are the coordinates of A then the coordinates of x are given by the formula:
X(A) = —(I + 32).
Putting this in equation (3.2) shows that (x(A)Je,Je) < 0, where V=span(e).

Thus x is transversal to £5 and defines a transversal orientation. 0

Consider a path A(t) which for |t| small lies in a single chart D, = DR, and

such that

A(0) 0 A0 = R60 , [60] = 1.

Let S,” = v(A(t)). Assume A(t) intersects £3, transversally at t=0. The transver-

sality can be rewritten as

(SfJeo, J60) aé 0.

where — here and below — the dot denotes g; at t = 0. Let M = {v 6 A0 ; [v] =

l, A(0) 6 Du} and define a map a = 0M.) : M —> {3:1} by

o(v) = sign(S,”Jeo, Jeo).

Lemma 3.4 For a path A(t) as above the map OM.) is constant.

18

Proof One can alternatively characterize M as {v 6 A0 ; Iv] = 1 , (v,eo) aé 0}.

Hence M has two components:
Mi = {v E M ;:t(v,eo) > 0}.

Now 5"," varies continuously with v, and obviously U(v) = o(—v). Thus a : M —)

{il} a continuous even map, so is constant. 0

Definition For a path A(t) as in Lemma 3.4 we define the local Maslov index by

p(Ao,A(t)) = oL(.)(v) , v E M. (3.3)

By Lemma 3.4 this definition is independent of coordinates.

We will next give several more concrete versions of formula (3.3). To begin,
note that in (3.3) oA(v) is independent of v, so we are free to choose v as we
please. Choose v = e0. Set f0 = Jeo and R, = \Ilo(15'1A(t)) where Io = 111,0.

(3.3) becomes:

MAO, A(t)) = sign (mfg, f0). (3.4)

Now consider the path

a. = f0 — Jnafa 6 151m).

Then 3:0 = f0 (since 60 E A(O) so that f0 = —I(‘,'leo E IJIA(O)) and hence

(iii-1‘0, 60) = ‘(JRtfo,€o) = (Rtf01f0)'

Differentiating at t=0 we get

(itaCO) = (Rtf09 fo)- (3.5)

19

Now introduce the conjugation D = D(t) associated to Io‘lA(t). Since 3; 6 151A“)
we have 3:, = D(t):ct. Differentiating this at t=0, taking the inner product with

co, and noting that D(0)eo = —eo we get
(530, 60) = (Dfo, 80) '- (i? 9 60)

and therefore

2950,60) = (Df0)60) = (JCfo, fo)- (3-6)

The conjugation associated with A(t) is C(t) = [51D(t)Io. Using this in (3.6) we
deduce

2(:i:o,eo) = (11001,-1 f0, f0) = (JCeo,e0). (3.7)

Combining (3.4), (3.5) and (3.7) we get:
Corollary 3.5
p(Ao, A(t)) = sign (JCeo , e0) = sign w(Ceo, e0)

where A(0) 0 A0 = Reo and w(:r,y) = (Jx,y) is the symplectic form.

Note that the above formula is independent of coordinates. For the application

we have in mind we will need another variant of this formula. Consider a family
U (t) E U] With
U(O) = I , C(t) = U(t)C(0)U(t)*.

If we write U = J A where A commutes with J and A is selfadjoint then

a = JAC(0) — C(O)JA = JAC(0) + JC(0)A

Ceo = JAC(0)eo + JC(0)Aeo = J(Aeo + C(0)Aeo) = J(I + C(0))Aeo.

20

But P(0) = 1/2(I + 0(0)) is the orthogonal projection onto A(O), so
(.1660 , Co) = -2(P(0)A€o , Co) = -2(A60 , 60).

Hence we have the following result.

Corollary 3.6 If A(t) = U(t)A(0) with U(t) = I + tJA + 0(t2) then

#(Ao, A(t))l,=0 = —sign (Aeo, co) = signw(er, co). (3.8)

Remark 3.7 There is an ambiguity in the definition of the Maslov index and
without a proper normalization the Maslov index is well defined up to a sign. This

is easily seen in the “mirror symmetry” of the Maslov index (cf. [CLMl], Prop.XI):

#(AIU), A2(t)) = _#(A2(t), A1(t))-

We consider as standard normalization the one in Property VII of [CLM 1] and we
want to compare it with our definition of Maslov index. For this we consider R2

with the standard symplectic structure
0 —1
w(x,y) = —(Jx,y) , J = [1 O [.

Let L0 = span(eo) where co = (1,0) and consider the path L: = eJ‘Lo fort in a

small neighborhood of 0. Corollary 3.6 gives
”(L0,Lt) = sign w(Jeo,eo) = 1

which agrees with the standard normalization.

Consider

7’0 = {’7 3 (I, 51) -* (Co,£5)} , I = [a,b] — compact interval

21

'p0‘ = {7 6 ’Po ; 7(t) intersects [.3 transversally }.

Since codim£3c = k(k + 1)/2 2 3 if k 2 2 we see that any path 7 in 730 can be

deformed (in 790) to a path in 190‘. For 7" 6 ’Po' define

#(A0a7.) = Z #(A017.(t) llt-t.‘|<e)°
‘Y'(ti)€£},

This is the usual definition of an intersection number. In particular standard
arguments show that the above u can be extended to the whole ’Po as a homotopy

invariant function. Now define
5(2). = {(211,112) 6 c”) ; A1 0 A1) = 0}
and
73(2) = {7: (I, 31) —-> (5(2) , 11(2),}.

Any 7 6 73(2) looks like 7(t) = (A1(t) , A2(t)). Without any loss of generality we
may assume that A1(0) = A0. We can find a smooth family of unitary operators
U(t) E 215 such that

A1(t) = U(t)Ao , U(O) = I.

Consider the family of paths 7, 6 13(2) defined by
7:“) = (U(s)‘1A1(t), U(3)'1A2(t))-

Define
#(7) = ”(Al(t), A2(t)) = ”(A0 a U(t)-1A2(t))'
We will check two things.

A. u(7) is independent of the family U(t) .

22
Indeed if U(t), V(t) e u,C are such that U(t)/lo -_- V(t)A0 then T(t) e 05(Ao). Set
A’ = U(1)-1A,(t) , A”(t) = V(t)-1Aa(t).
If A’(t) = T(s)A’(T) then (A0, A‘(t)) 6 5(2)) and A°(t) = A’(t), A1(t) = A”(t) i.e.
#(Aoa A’(t)) = #(Ao, A”(0)

and this proves A.
B. u(7) is a homotopy invariant. The proof is entirely similar to the proof of A.
In fact both rely on the fact that the inclusion

£0 ‘—+ £(2)

is a homotopy equivalence. The details are left to the reader .

An immediate consequence of the above considerations is that [1 defines a morphism

u : 7r1(£(2)) H Z.

The finite dimensional Maslov index behaves nicely with respect to symplectic
reductions. So does this infinite dimensional version of it. Recall first the process

of reduction.

Lemma 3.8 Consider A C H a lagrangian of H, an isotropic subspace W and its
annihilator W0. If (A, W0) is a Fredholm pair then:

(i) H0 2 Wo/W has an induced symplectic structure;

(ii) AW = (A O W0)/W is a lagrangian subspace in W/WO.

Proof (i) is straightforward and is left to the reader. We now prove (ii) in a

special case which is precisely the situation we will ever need. We will assume that

23

A is clean mod W i.e. A 0 W = 0. We will identify Hg with the orthogonal

complement of W in W”. Finally set U = (W°)J*. Then

Denote by P0 (resp Pw,PU ) the orthogonal projections onto 'Ho (resp. W,U ).
Obviously AW is an isotropic subspace of Ho. We show it is maximal isotropic.

Let ha 6 Ho such that (Jho,AW) = 0. Then
Jho 1 (AW + W) => Jho 1 (A n W)
i.e.
ho e J(A n W0)l = J(Ai + U) (since (A,w°) is Fredholm)
-_- J(JA + U) = A + W (since Wis isotropic).

Thus ho 6 HO O (A + W) i.e. ho 6 AW. Lemma is proved. O

For any isotropic subspace W, JW is also isotropic and we define
£(2)W(H) = {(A1,A2) e c”) / (A1, W) is Fredholm A1 n W = A2 n JW = 0}.

(The pairs of 5(2)“, are called clean mod W). Note that if (A1,A2) 6 U2)”,
then (A2,(JW)°)) is Fredholm and we have a natural identification WO/W a
(J W)°/ .I W (given by J). The reduction process described in lemma 3.9 induces a
map

«W : £<2>W(H) —» 5mm.) (A1,A2) H (AK/1gp”).

Since the reduction is clean mod W we deduce as in the finite dimensional case
that Mix is continuous (see [GS]). As in finite dimensions we have the following

result.

24

Proposition 3.9 (Invariance under clean reductions) If 7(t) 6 P”) is clean

mod W at any time then

Proof: As before it suffices to consider only the special case 7(t) = (A0 , A1(t))

where t is very small. We can assume without any loss of generality that
A10) = U(t)/M0) , U(O) = 1 , U(1‘)le E 1-

Let U(O) = JA. Clearly A _=_ 0 on JW. Using (3.10) to compute the local Maslov
index we see that W has no contribution in the formula and thus nothing changes

if we mod W out. 0

Remark 3.10 One can show that the map 7rw is actually a homotopy equivalence.
A similar result holds if we allow W to vary with t. As long as the reductions stay
clean we have the invariance of the Maslov class (see [DP], [V] for a related result).

We leave the details to the reader.

Using the homotopy long exact sequence for the pair (U2), (2(3)) and the results

proved so far we deduce
Theorem 3.11 Let 7o , 71 6 79(2). Then 70 is homotopic to 71 if and only if

#(70) = ”(’71)-

In particular ,u : 7r1(£(2)) —> Z is an isomorphism.
The details are left to the reader.

We now have a flexible definition of the Maslov index. In the following chapters

we will apply it in connection to spectral flow computations.

Chapter 4

Boundary value problems for
selfadjoint Dirac operators

We gather in this chapter various analytical facts about boundary value prob-
lems for Dirac operators. Many of these results are known (see [BW4]) but we

reformulate them in a form suitable to our purposes.

Consider an oriented Riemann manifold (M,g) and 8 —> M an euclidian vector
bundle over M. Denote by C(M) the bundle of Clifford algebras over M whose fiber
at x E M is the Clifford algebra C(TgM). We will assume that E is a selfadjoint
C(M)-module that is for each 1-form 77 E 91(M) the Clifford multiplication 0(7)) 6

End(£) is skew-adjoint.

Definition 4.1 ([BGV], Chap.3) Let D : C°°(5) —> C°°(8) be a Ist order differ-

ential operator. Then D is called a Dirac operator ifV f E C°°(M)

C(df)u = ID, f]u = D(fu) - f(DU) Va 6 C”(5%

In the sequel all Dirac operators will be assumed formally selfadjoint.

It is easy to construct Dirac operators. Let V be the extension of the Levi-

Civita connection to C( M). Fix a connection V5 on 8 compatible with the Clifford

25

26

multiplication in the sense that Vc = 0, that is
V(C(U)) = C((VUD- (4-1)
Then the composition
C°°(8) 2i C°°(T‘M®£) 5.. C°°(e)
is a Dirac operator D = D(V'S). In dual local framings
D = Z C(e")vfi.

The space of selfadjoint Dirac operators compatible with a given Clifford action is

an affine space modelled on the space of symmetric endomorphisms of 8.

 

 

Zx{-l} 23({0}

Figure 4.1: The metric is cylindrical in a neighborhood of the boundary

Let M be a compact oriented manifold with boundary 2 = 8M and suppose
it is endowed with a cylindrical metric in a neighborhood of the boundary. More
precisely if U C M is a collar neighborhood of E in M with an identification
w : U E 5.3 X (-1, 0] then in these coordinates the riemannian metric on M satisfies
9],, = h + ds2 where h is a riemannian metric on E (Fig. 4.1). Denote by V = V9

the corresponding Levi-Civita connection. Let 8 be a selfadjoint C(M)-module and

27

V a Clifford connection on 8. Set 80 = 5|E and pick an isomorphism (cylindrical

gauge)
‘1’ I ElU E“ 80 X (—1,0]

covering it) such that over the neck
v=v°+ds®ams

where V0: VIE. Fix once and for all the isomorphisms it) , \II, the connection V

and the metric g.

Definition 4.2 A Dirac operator D is called cylindrical if over U it has the form
D = c(ds)(8/Bs + D0) (4.2)

where Do : C°°(£0) —> C°°(€o) is independent ofs over U'. In addition if D0 is

selfadjoint then D is called neck-compatible (n.c.).

In the sequel all Dirac operators on manifolds with boundary will be assumed

cylindrical.

For example D = D(V) is a n.c. operator. To see this consider e1, 62, - - - , en, 0/33

a local orthonormal frame in U (n = dim 2). Then over the neck U
1‘) = C(da)(a/aa + D.)

where

Do“ - 2C0?) C()dS V0 (4.3)

is independent of s and selfadjoint because c(e )c (ds) 1S skewadjoint and commutes

with V: .

28

Remark 4.3 Do is a Dirac operator over 2. To see this fix :1: E E and set V0 2
ET], V = Vo€B(ds) E T;M , E0 = £0lx- Let 61, e2, - . - , e" be an orthonormalframe
of V6. E0 is a C(V)- module. It inherits a structure of selfadjoint C(Vo)-module
induced by the embedding of Clifford algebras

C(Vo) HC(V), eIH elods

(here I is an ordered multiindex: 1 S i1 < moi), _<_ n) Denote by Co the action
of C(Vo) on E0 so that c0(eI) = c(eI)c(ds). In particular we deduce that V0 is a
Cliflord connection compatible with the above Clifford action and D0 = D0(V°) is

the Dirac operator associated to V0.

If A is a cylindrical endomorphism of 8 i.e. a selfadjoint endomorphisms
satisfying B/BsA = 0 over U then D + A is cylindrical. We deduce that the space
of cylindrical Dirac operators is an affine space modelled on the space of cylindrical

endomorphisms.

If A is a neck compatible endomorphism of E i.e. a cylindrical endomorphism
anticommuting with d(ds)

{A, c(ds)} = 0 (4.4)

over U then D = D + A is a n.c. Dirac operator. Indeed it is cylindrical and if we
set A0 = AIS , BO 2 Aoc(ds) then we deduce that Bo is selfadjoint since A0 is

selfadjoint, C(dS) is skewadjoint and A0 and C(ds) anticommute. Over U

D = C(dS)(G/GS + Do) , Do = D0 + Bo.

We next turn to the analytic aspects of cylindrical Dirac operators. The ad-

equate functional framework for all our future considerations is that of Sobolev

29

spaces L3 (functions “o-times differentiable” with derivatives in L2). We will de-

note the norm of L: by I I0 and the L2 norm by I I.

Let D be a Dirac operator. Following Seeley [S] we consider the spaces
K(D) = {u E C°°(E) ; Du = 0;in M}

MD) = IC(D)nL§

Since D is elliptic Ka(D) C C °°(£' ) We are interested in the subspace spanned
by the restrictions over 2 of the sections in K0(D). For a > 1/2 the existence of
these restrictions is a consequence of classical trace results for Sobolev spaces (cf.
[LiMa]). The case a = 1/2 requires a more subtle treatment since the usual trace
map is not defined. One uses the fact that K1/2(D) is a distinguished subspace of
sections satisfying an elliptic PDE and a growth condition near the boundary. For

3 6 (0,1) consider the restriction map
12. = C°°<£> —> Gust) u H “I2x{s}
For any u E K1/2(D) the limit (in L2(80))
Ron d2 [13612.21

exists and is uniform in {lull/2 < 1} fl K1/2 (see [BW4], [8]). This limit map has

two important properties.

Proposition 4.4 R0 : [Cl/2 —+ L2(8o) is a continuous map satisfying
(a)Unique continuation: Ifu E [Cl/2(0) and R0(u) = 0 then u=0.

{b)Boundary estimates: If u E K1/2(D) then

lull/2 S const.IR0uI

30

For the proof we refer to [BW4] or [8].

Definition 4.5 The Cauchy-data space of D (CD space) is the subspace A(D) C
L2(80) defined as
A(D) = Bowl/2(1)» = ’CI/2(D)|2-

One sees that A(D) is a closed subspace of L2(€0). It is roughly the subspace
consisting of those sections u E L2(80) which extend to a solution of DU = 0 over
M. Proposition 4.4 shows that R0 is a linear isomorphism between K1/2(D) and
A(D). The orthogonal projection P(D) onto A(D) is usually called the Caldéron
projector of D. By the classical results of [8] this projection is induced by a

0th order pseudodifferential operator whose symbol can be explicitly computed

([BW4], [P2], [5])-

The dependence of the Caldéron projector on the operator is rather nice.
The method of constructing the Caldéron projectors detailed in [BW4], Theorem

12.4(b) (see also [8]) can be used to prove the following result.

Proposition 4.6 Let {D‘} be a family of cylindrical Dirac operators on M com-
patible with a fixed Cliflord action. Assume Dt is smooth in some Sobolev norm
Li, where k is sufliciently large so that L], ‘—-) C2, (e.g. k 2 N/2+2, N = dim M).
Then the path of orthogonal projections II. onto A(t) = A(D(t)) is C1 as a path in

the Banach space of bounded operators L2(£o) -—> L2(€o).

Proof We begin by briefly recalling the construction of the Caldéron projection.

Let M denote the double of M: M = M U; (—M). Continue to denote by s
the longitudinal coordinate along a tubular neighborhood N of E in M so that
NEEX(—1,1)and3£00nM.

31

For each Dirac operator D over M denote by D = D(D) the invertible double
of D constructed in Thm.9.1 of [BW4]. This is an invertible Dirac operator on a

bundle 8 over M, extending D. Moreover D depends smoothly upon D.

For every u E C°°(80) denote by 6 (8) u the vector-valued distribution over M
defined by
(6 at, V) = [,an )2) V e o°°(é).

Note that supp 6 8) u C E and 6 ® u E LEI/2% for 0 < 5 < 1/2. This follows from

an equivalent description of the map u H 6 (8) u as the adjoint of the trace map
7 : C°°(é) —> C°°(80) V H v (a.

This adjoint is a continuous operator 7“ : L30 —) L2—1/2—a for all a > 0.

Given u E C 00(80) denote by U = U (u) the distribution over M defined by
U = D‘1(6 (8) u). By classical regularity results sing supp U C E and U 6 L324.
In particular U is smooth over the interior of M and in [BW4; Thm12.4] or [8] it
is shown that
RSU = slim—U I E x {3}

exists in any C k norm. The basic result is that

H(D)u = RgU Vu e C°°(£o).

Now let D‘ be a smooth path of Dirac operators over M and set 11, = U(D‘),
D, = D(D‘) and let II - II denote the natural norm in the space of bounded linear
operators L280) —» L2(80). One fact that will be frequently used in the sequel is

the following inequality for distributions over a compact manifold.

[at], _<_ C(a)||a||ca|¢|, 0 s o g 2, a e 02, 45 6 L3,. (4.5)

32

The proof of (4.5) is immediate. For a = 0 or a = 2 this is simply the Holder

inequality. For the other a it follows by interpolation.

The proof that t H II, is smooth is carried out in several steps. For every

u e C°°(£o) and any t set U, = 13,-‘(6 <8) u). Fix a e (0,1/2). Note that
IUtI1/2-e S CIUI (“D
where C > 0 is independent of t at least for all small t.
Step1 We will prove that U, — U0 E L3,“ and
IU, — U0I3/2_, S CtIuI Vu E C°°(6'0) (4.7)
where C > 0 is independent of t (small).
To prove (4.7) write D, as D, = D0 + A(t) where A(t) e End(é) satisfies
Hume=OMauea as
U, satisfies the equation DOU, + A(t)U, = 6 (8) u so that
D0(U, —- U0) = —A(t)U, E Lf/2_€.

By standard elliptic regularity we deduce U, — U0 E L§/2_5. Using elliptic estimates

and the invertibility of Do we deduce

la—asm<cmmwma

The estimate (4.7) follows immediately using (4.5), (4.6) and (4.8).

Step 2
”II, — HOII = O(t) as t H 0. (4.9)

For u E C°°(80) we have

II,u = RgU, = RgUo+Rg(U.—Ua) = nau+R;(U,—Ua).

33

The existence of R5 (U, — U0) follows from the regularity established at Step 1 and

classical trace results. In particular
IHgU — HoUI = IRS(U¢ — U0)I _<_ CIU; — U0I3/2_5.

The estimate (4.9) now follows from (4.7). In particular we proved that If, depends

continuously upon t.

For any u E C°°(€o) let V0 be defined as the unique solution of the equation
Dove + A(0)U0 = 0 (4.10)

where as before D, 2 D0 + A(t), U0 2 D51(6 (8) u) and the dot denotes the

differentiation at t=0.

Step 3

IU, — U0 — tVoI3/2_, g Ct2IuI for all u e C°°(£o) and all t small. (4.11)
To prove (4.11) write A(t) = tA(0) + B(t) where R(t) E End(f:') and

IIR(t)I|02 = 0(t2) as t —+ 0. (4.12)
The equation D, U, = 6 (8) u can be rewritten as
DOU, + tA(0)U, + R(t)U, = 6 a u.
Using (4.10) and DOUG = 6 <8) u we deduce
vow. — U0 — tVo) = —tA(0)(U, — U0) — R(t)U,.

Hence by elliptic estimates we have

an - Ua — tVoIs/z—e s C (tlA(0)(U. - U0)I1/2-e +|R(t)Ut|1/2_.).

34
The estimate (4.11) follows easily using (4.5), (4.7) and (4.12).
Coupling elliptic estimates in (4.10) with the relations (4.5) and (4.6) we get
IVOI3/2—e S CIUI (4-13)
The reader can now verify immediately that t —) II, is C1 and
not = 1231/,

Proposition 4.6 is proved. 0

We can now relate the Dirac operators and their CD spaces to the infinite
dimensional symplectic topology of the previous chapters. All this setup lies over
a natural symplectic background. Indeed C(ds) is a fiberwise isometry so it defines

a unitary operator

J I L2(£0) —+ L2((€0) , J2 = -I

i.e. J is a complex structure on L2(80) thus defining a symplectic structure

w(u,v) = [2(Ju,v)
for all u,v E L2(80).

The next result is the key fact which unifies all the topics discussed so far. It is

another manifestation of the duality 5' el f adj oint operators ~ Lagrangian subspaces.

Proposition 4.7 ([8 W2], Prop.3.2) A(D) is a lagrangian subspace of L2(£0) with
respect to the natural symplectic structure induced by the Cliflord multiplication

with ds.

Finally consider the following situation. (M,g) is a compact oriented Rie-

mann manifold and E a selfadjoint C(M)-module over M. Let 2 be an oriented

 

 

 

 

 

Figure 4.2: A metrically nice splitting

hypersurface in M which divides it into two manifolds-with-boundary M1, M2.
Choose N1 , N2 tubular neighborhoods of M1 , M2 such that N1 E E X (—1,0],
N2 E E x [0, 1) (see Fig. 4.2). Set N = N1 U N2. We assume the metric g is a
product metric on N i.e. gIN = ds2 + h, where h is a metric on E and —l < s < 1
is the longitudinal coordinate on N. Let D : C 00(6 ) —> C 0"(6') be a Dirac oper-
ator on M. Denote by D, (resp. D2) its restrictions to M1 (resp. M2). D will
be called cylindrical if both D1 and D2 are cylindrical. As usual set 80 = EIE.
L2(£o) has a symplectic structure induced by the Clifford action. The CD spaces
A1(D) , A2(D) of D1 and D2 are lagrangian subspaces in L2(50). In fact more can
be said.

Proposition 4.8 (A1(D), A2(D)) is a Fredholm pair. Moreover (A1, A2) is a transver-

sal pair if and only if D is invertible.

Proof Let P,- be the orthogonal projection onto Aj, j=1,2. We have seen that

these are 0-th order pseudodifferential operators in L2(80). In [8] (see also [P2],

36

Chap.XVII) it is proved that their symbols satisfy

wills) + U(P2)(€) = Id.

Thus P1 — (I — P2) is a pseudodifferential operator of order S —1 in L2(50). In
particular P1 — (I — P2) is compact so that (Al, A2) is a Fredholm pair. The second

part is intuitively clear (see also [BW2] Corollary 3.4). O

Chapter 5

The Maslov index and the
spectral flow

The setting of this chapter is identical to the one at the end of Chapter 4. We
endow the space of cylindrical Dirac operators D with a Sobolev topology, given

by a LI“: norm with k sufficiently large so that L: H C2. Inside D sits
D“ = {D E D ; D is invertible}

To any continuous path 7 = D(t) in D with endpoints in D“ one can associate
an integer, the spectral flow SF (7) (see [APS3], [BW1]) defined as the number
of eigenvalues of D(t) that change from negative to positive minus the number of
eigenvalues that change from positive to negative. This is a homotopy invariant of 7
(cf. [AS], [BW1]) with an obvious additivity property. If 71, 72 : (I, BI) —-> (D, D‘)

SFI71'72) = 5F(’71)+ 517(72)-
so the spectral flow can be viewed as a homomorphism

SF: 7r1(D,D*) _. z.
37

38

In the previous chapter have defined a continuous map
A”) : (D, 13*) a (£(2),££2)) , D H (111(1)), 112(0)).

Denote by A?) the homomorphism between nl’s induced by this map.

In Chapter 3 we constructed the Maslov index isomorphism
u : 7rl(£(2),£(,2)) -—-> Z.

We will prove that the following diagram is commutative.

 

(2)
71(D7DI) A. : 7r](£(2)1£'('2))

SF u

To this end we will need a localization procedure for the spectral flow. Let
t H D(t) E D (ItI S e ) be a smooth family of cylindrical Dirac operators such
that D(t) is invertible for t # 0. Let K0 = kerD(0) and denote by Po the

orthogonal projection onto K0. We form the resonance matrix:

R = B(A) :Ko ——> K0 R = P0D(0)
We can view R as a symmetric matrix. We have the following result ([DRSI).

Theorem 5.1 Let D and A as above satisfying (1). If the resonance matrix R( A )

is nondegenerate then its signature gives the spectral flow

SF(D(t) ; [t] S e) = sign B(A)

39

The above formula follows from an abstract result of Kato (Thm.II.5.4 and 11.6.8
of [K] )which we recall now. H is a separable Hilbert space and A(t) t E R
a family of unbounded selfadjoint operators with a fixed dense domain W. W
becomes a Hilbert space in its own right using the graph norms. We assume that
the embedding W H H is compact and that the resolvent set of A(t) is nonempty
for every t. Then A(t) has compact resolvent and its spectrum consists entirely
of eigenvalues with finite multiplicities. A(t) can also be interpreted as bounded
operators W —> H. As such we assume that A(t) depends smoothly upon t. The

following result gives a precise information about how the eigenvalues of A vary.

Theorem 5.2 (Kato Selection Theorem) Let to E R and co > 0 such that
:tco E 0(A(to)). Then there exists a constant e > 0 and differentiable functions
A,- : (to—€,to+€) --> (—-co,c0), j = 1,2, - - - ,N {N is the dimension of the subspace
spanned by the eigenvectors corresponding to eigenvalues in (—-c0,c0)) such that
/\,~(t) E o(A(t)) and
Xi“) E U(Pj(t))4(t)Pa(t)

where Pj(t) : H —> H denotes the orthogonal projection onto her (/\,(t)I — A(t)).
Moreover if /\ E o(A(t)) fl (co,co) with corresponding spectral projection P : H —)
ker (AI—A(t)) and 0 E o(PA(t)P is an eigenvalue of multiplicity m then there are

precisely m indicesj1,- - ' ,jm such that Ajy(t) = A and Xju(t) = 9 for V = 1, - - - ,m.

Kato Selection Theorem has a corollary particularly important for our purposes.

To formulate it introduce the set of positive cylindrical endomorphisms
Cyl(€)+ = {A e Cyl(£) /31 > 0 : info(A(:L‘)) 2 A Vx e M}
where o(A(x)) is the spectrum of the selfadjoint endomorphism

A(x) : 8,, —l 5,.

40

Set

7: = (741,01)... (19,77) ;7 6 01}-

A path 7 E 'P is called positive if 7 E Cyl(£)+ and negative if —7 E Cyl(£)+.
The set of positive (resp. negative ) paths is denoted by 73+ (resp. P-) The

resonance set Z = Z(7) of a path 7 E 'P is defined as
Z = {t E I ; kerD(t) ;é 0}.
We can now formulate

Lemma 5.3 The resonance set ofa positive path is finite.

Proof Let 7 = D(t) E ’P+ and to E Z(7). Since D(to) E Cyl(t‘:)+ the resonance
matrix is positive definite and by Kato selection theorem we deduce that D(t) is

invertible when t is in some e-neighborhood of to. Therefore Z (7) is a discrete set.

0

Positive paths have other important properties.

Lemma 5.4 Any path 7 E P is homotopic to a product of a positive path with
a negative path. (In the sequel all the homotopies of paths (1,81) —+ (D, D‘)
will be understood as relative homotopies - the endpoints stay invertible during the

deformation).

Proof The difference A = D(l) — D(O) E Cyl(8) is a bounded selfadjoint endo-

morphism of 8. Choose C > 0 such that
C 2 1+ Isupo(A(x))I Vx E M (5.1)

mm+ovaen. an

41
The choice (5.2) is possible by Lemma 5.3. Now consider
(1+ = D(O) +tC-1d5 tEI
a- = D(O) + C-Idg + t(A — C'Idg) tEI.
By (5.1) and (5.2) oi E Pi. 7 is homotopic to (1+ - 0. via an affine homotopy. 0
Definition 5.5 A C1 path 7: (1,81) —> (D, D‘) , t H D(t) is called

(i)locally affine if7 = const. in a neighborhood of any t E Z(7);

(ii) standard if Z(7) is finite and Vt E Z(7) dim ker D(t) = 1.

A key step in our deformation program is a genericity result which states that

almost any path of Dirac operators is standard.

Proposition 5.6 Let D be a cylindrical Dirac operator and assume rank? 2 2.
Then there exists a Baire set Ares C A such that for A E Ares the path D(t) = D

+ A(t) is standard.

The proof of this proposition is carried out in the Appendix.

In particular since P+ is open in P we deduce
Corollary 5.7 Any positive path is homotopic to a positive standard path.
A simple application of Kato’s selection theorem yields

Lemma 5.8 A positive standard path 7 E P is homotopic to a locally afine posi-

tive standard path 7 such that:
(i) Z(7) = Z(7)

WWE Z(7) = 7(t)=’7(t)-

42

Proof The underlying idea is natural: any path is locally homotopic to the tangent

line at a point on the path. The only thing we have to prove is that we can find a

relative homotopy achieving this. Assume 7 : [—1,1] —+ D an (1 Z (7) = {0}. Set

D(t) =D(0) + A(t) and A0 = A(O). A0 is a positive cylindrical endomorphism of

5. Consider

~

D,(t) = (1 —s)D(t) + stAo s e [0, 1]

By Kato Selection Theorem there exists 5 > 0 such that V 0 < ItI S e D(t) is

invertible and its inverse E (t) satisfies

1
“E(t)” = 0 (m)

Now

alt) = D(t) + Raul
where R,(t) = s(tAo — A(t)) satisfies
IIR,(t)II = 0(t) uniformly ins
Thus
E(t)D,(t) = I + K,(t) K,(t) = E(t)R,(t)

where (by (5.3))

IIK,(t)II = 0(1) uniformly ins

Hence we can find to > 0 such that

 

 

K,(ito)|| <1/2 V3 6 [0,1]

(5.5)

(5.6)

and from (5.5) we deduce that D,(:l:to) is invertible for all 5. Therefore D,(t) is

an admissible homotopy between 7 and a locally affine path satisfying properties

(i) and (ii) in the lemma. 0

43

The homotopies constructed so far were between paths close to each other in
the C1 distance. Our next result describes one instance of homotopic paths which

can be Cl-far apart (but still C()-close).

Lemma 5.9 Let D E D and A E Cyl(€) such that dim ker D = I,

ker D = span(U)

and

(AU,U) as o.

IfB E Cyl(€) is such that

(BU,U) = (AU,U)

then 38 >0 such thatV0< ItI Se anstE 1

73(t) = D + (1 — s)(tA) + s(tB) E D'.

In particular 7,(-) E P realizes an affine homotopy between 70(t) = D + tA and
71(t) = D + tB (ItI S 6).

Proof The paths 7,(t) are analytic in t (being affine). In such situations more
powerful perturbation results are available. In particular by Thm.VII 3.9 of [K]

there exists 61 > 0 and analytic functions

Am, : [—el,el] —> R n E Z s E [0, 1]

such that

o(7,(t)) = {A,,,,(t) / n E Z} (multiplicities included).

44

We labelled the eigenvalues so that A0,,(0) = O E kerD. Note that An,,(0) is
independent of s for all n E Z. We will denote it by An. On the other hand we can
find a,b > 0 such that Vv E C°°(8)

||7s(t)v|| S. allvll + b||7,(t)v|| Vltl S 6 V8 6 [0,1]-
Theorem VII 3.6 of [K] implies
IAn,s(t) - An] S C(1+ Anlt
where C = C(a, b) > 0 is independent of n E Z and s E [0,1]. In particular for

o < [til 3 52 = ini[2c(l':‘(l‘::'(0)l) /n e Z\{O}}U{el}

 

IAa,a(t)| 2 1/2lln(0)| > 0- (5-7)

On the other hand by Kato Selection Theorem
10,,(0) -_-. (AU,U) aé o VsE [0,1].
Arguing by contradiction we can find 0 < e < 52 such that
A0,,(ie) # 0. (5.8)

In particular (5.7) and (5.8) show that the operators 7,(:te) are invertible for any

3 and Lemma 5.9 is proved. 0

Definition 5.10 A standard path 7 E P is called elementary 2'th E Z(7)
7(1) 2 afdg

for some a E C3°(M) a function supported in M2 \ N and not changing sign (see
Fig. 5.1).

45

 

 

 

zit-1} me} Exil}

Figure 5.1: The cutoff function 0

Remark 5.11 [fa E C8°(M) is as in Definition 5.2 and D E D then the unique

continuation principle for Dirac operators ([BW4], Chap.8) implies that

(aU,U) 75 0 VU E kerD \ {0}.

Lemmata 5.3, 5.4, 5.8, 5.9, Corollary 5.7 and Remark 5.11 have the following

corollary.
Corollary 5.12 Any path 7 E P is homotopic to an elementary path.
In particular we have the following abstract result.

Proposition 5.13 Let qt : P H Z be a continuous, additive function such

that for any elementary standard path to

and Qty) = O for every 7:] H D”. Then V7 6 P .‘ (25(7) = SF(’7)

Let 7 E P , 7(t) = D(t). Denote by D,(t) the restriction of D(t) to M,- j=1,2.
Let Aj(t) be the CD space of Dj(t) j=1,2. Since D(O) and D(l) are invertible

46

we deduce A, (t) D A2(t) = 0 for t=0,1. The results of section 4 show that the
Fredholm pairs of lagrangians (A1(t) , A2(t)) vary smoothly with t. In particular
the Maslov index u(A1(t) , A2(t)) is well defined. We can now state the main result

of this paper
Theorem 5.14 For any path 7 E P as above we have

SFU) = H(A1(t)1A2(t)) (59)
Proof We have defined a map (15 : P H Z

t = 7 = D(t) H # (A1(D(t)), A2(D(t)))-

By Propositions 4.6 and 4.8 we see that d is continuous and <15 = 0 on the paths in
D‘. By Proposition 5.13 it suffices to check (5.9) on elementary paths. Thus fix a

cylindrical Dirac operator such that
kerD = span(Fo) , IFOI = 1

and consider the family D(t) = D + to] with It] S e where a is a smooth not-
changing-sign function ,compactly supported inside M2, away from the neck N.

The operator D1(t) is not changing since a is supported outside M1. Thus
mm “‘2 A0
is constant and A(t) d—i—f A2(t) is varying.
Let U(t) be a smooth path of unitary operators on L2(Eo) such that
U(O) = 1 , A(t) = U(t)A(0).

Set f0 = RFD , f, = U (tlfo be the restriction of F0 to 2 (we adopt the convention

of using capital letters for sections of 8 defined over M, M1 or M2 and small letters

47

for sections of 8 defined only over 2). Then f, lies in A(t) so there exists an unique

F, E ker D2(t) such that

RF, 2 U,f0 on E

U, f0 varies smoothly with t and the boundary estimates of Prop.4.4 imply that F,
depends smoothly upon t as well. Derivating (5.10) at t: 0 (the dot will denote

the t-derivative at t=0) and noting that D2 = 01 we get

D,(0)FO + aFo = 0_ in M2
RFD = Ufo on 2

Multiplying by F0 we get
—(aFa, F.) = (Damn, Fa).
Now if we integrate by parts in the above equality and use (5.10) we obtain

/M2<D2(0)F0, F0) = —/E<JF0, F0) + /M2<FO, D2(0)F0) : —(']Uf03 f0)

Thus
(OFO. F0) = (JU(0)fo 1 f0) = W(Uf0 t fol- (5-11)
By unique continuation (aFo , F0) 7:4 0. The sign of the left hand side of (5.11) is

equal to SF(D(t); ItI S e) by Theorem 5.1. The sign of the right hand side is equal

to the Maslov index u(A, A(t)) by Corollary 3.6. This completes the proof. 0

Chapter 6

Adiabatic limits of CD spaces

Consider a manifold with boundary M as in Chapter 4 and D a neck-compatible
Dirac operator on M. Define M (r) = M U E x [0,r]. M(r) is usually called an
adiabatic deformation of M; (see Fig. 6.1). D has a natural extension D(r) as a
neck-compatible Dirac on M(r) and denote by A’ C L2(£ I3M(,)) the CD space of
I)(r).

In this chapter we will study the behavior of A" as r H 00. On the tube
2 x [0, 00) the operator D has the cylindrical form D = c(ds)(3/as + Do) so that

“DOTAO i.e. we are dealing with a dynamics

at least formally we may write A’" = e
problem on a lagrangian grassmanian. From this representation we see that the
part of A0 “interacting” with the negative spectrum of D0 will have a dominant
effect as r H 00 while we expect that the “interactions” with the positive spectrum
will “soften” as r increases. We may continue our formal discussion by observing
that since Do anticommutes with J it lies in the ”Lie algebra” of the infinite

“00" is a l-parameter group of

dimensional symplectic group so that the “flow” 6
symplectic transformations of H and the family AT is a trajectory in an infinite
dimensional lagrangian grassmanian. Unfortunately these observations are purely

formal since Do cannot generate a semigroup (the spectrum is unbounded both

48

 

 

glue

Figure 6.1: Adiabatic deformation of the neck

from bellow and from above). However, in finite dimensions this discussion makes
sense and the first result of this section, Proposition 6.3, describes the asymptotics
of this flow. The study of the infinite dimensional situation will ultimately rely on

this result via a careful symplectic reduction.

Since we will be dealing with asymptotics of families of subspaces it is appropri—
ate to begin our presentation by discussing ways to measure the distance between
two closed subspaces in a Hilbert space. The right notion is provided by the gap

distance between two subspaces introduced in [K].
Let X,Y be two closed subspaces of a Hilbert space H. Define
6(X,Y) = sup{dist(x,Y) ; x E X IxI =1}.

6 is in general not symmetric in X and Y. We symmetrize it by defining the gap

between X and Y as
6(X,Y) = max{6(X,Y),6(Y,X)}.
Note that 6(X, Y) can also be characterized as the smallest number 6 such that

dist (x,Y) S 6le V x E X.

50

We say Xu H X if 6(Xn, X) H 0. In particular if Pn are the orthogonal projections
onto Xn then

XnHX <2? PnHP innorm.

Thus if H is symplectic the gap topology on the space E of lagrangians is equiva-
lent with the natural topology (defined by the identification (2.2)). Although the
function 6(. , .) is in general not symmetric in its arguments it becomes symmetric

when restricted to .C. Indeed by Thm. IV.2.9 of [K] we have
6(L1,L2) = 6(L2i,L1i).
Since L1 , L2 are lagragians
6(L5L,Lf‘) = 6(JL2,JL1) = 6(L2,L1).
At the last step we have used the fact that J is an isometry. Thus

Ln H L... in .C 4:) 6(L,,,L..) H 0.

In studying convergence of sequences of subspaces it is very convenient to have a
method to “renormalize” them (much like the homogeneous coordinates in the pro-
jective spaces). We can achieve this if we can represent these subspaces as graphs of
linear operators. This representation is possible once some obvious transversality
conditions are assumed (compare with Arnold’s charts on lagrangian grassmani-
ans). When these renormalizations are possible there are ways to relate the gap
topology with the norm topology of linear operators. In particular we will fre-

quently use the following results. Their proofs can be found in [K].

Lemma 6.1 Let H1 and H; be two separable Hilbert spaces and consider a se-
quence (Tn) of bounded linear operators Tn : H, H H, with graphs G(T,,) C

H, {B H2. Then the following are equivalent

51
(i) Tn H T as n H 00 in norm;

(ii)G(T,,) H G(T) as n H 00 in gap.

Now consider H = R2" with the complex structure J induced by the identifi-
cation R2” E C”. J defines a symplectic structure to by w(x, y) = (Jx, y) for all

x, y E H. The symplectic group is then
Sp(n,R) = {T E GL(2n,R) / T‘JT = I}.
Sp(n,R) is a Lie group with Lie algebra
E(n,R) = {A E g](2n,R) / A‘J+ JA 2 0}.
Inside £011 R) sits the subspace
aln) = {A e £13(naR)/A = A*}

consisting of selfadjoint matrices anticommuting with J. Denote by A(n) the la-
grangian grassmanian of (R2",J). Sp(n,R) acts (transitively) on A(n). In par-
ticular any A E U(Tt) defines a 1-parameter group of diffeomorphisms of A(n) :
r H 6"”. The problem we intend to discuss is that of the asymptotic behavior of

the above flow on A(n). Fix A E U(n) and consider
1,, = {LEA(n)/ALCL}

the family of invariant lagrangians of A. The lagrangians in 1A are stationary points
of the flow r H e’”.

Let us now describe the dynamics of e‘”A in a simple but instructive case.

Example 6.2 Take n=1 and fix A E o(l)\{0}. We can then choose e E R2 , [cl =

1 such that in the basis (e,Je) the operator A has the form A = diag (A , —A ).

 

 

I)

Figure 6.2: Hyperbolic flow

Figure 6.3: Dynamics on A(1)

Viewed as a (linear) flow on R2 e‘” has the hyperbolic phase portrait depicted
in Fig. 6.2. The lagrangians of R2 are the lines through the origin so that A(1) E
RP1 E 51. e‘TA becomes diag(e”\", e“). H- = span(f) and H+ = span(e) are

the only stationary points of the flow. If L ;é H+ then one sees from Fig. 6.2 that
e’”AL H H- exponentially as r H 00.

The phase portrait of e“TA on A(1) is then the one described in Fig. 6.3. In

particular we have shown that VL E A(1) e’rAL has a limit in 1,, as r H 00. O

The situation presented in the example above is a manifestation of a more general

53

phenomenon.

Proposition 6.3 Let a be a real n x n symmetric matrix. Then for any subspace

U C R” there exists U00 - an invariant subspace of A such that

lim e-IAU = U00

T—’OO

Proof Assume U(A) = {A, S S An} with the corresponding orthonormal

spectral basis e1, - - - ,en.Pick u1,- - - ,um (m = dim U ) a basis of U. Then

n
u,=ZC,-jej i=l,---,m
1:1

and we can form the matrix

We may assume C is upper triangular. Otherwise we can reduce it to this form
by performing row operations (which is equivalent to choosing a different basis for
U). For each 1 S i S m let j(i) be the smallest j such that C,‘j # 0. Since C is

upper triangular

 

j(l) < < j(m)- (6-1)
For r 2 0 let
v,-(r) = 1 e"\1(')e_”Au,°.
Ciao“)
v1(r), - - - , vm(r) form a basis of e’IAU and moreover
'U,‘(OO) = rlil'glo v,(r)

exists and for all i and

v,(oo) = 61‘“) + Z ugkek.
k>j(i)

54

From (6.1) we deduce that v,(oo) are linearly independent and therefore
e"’AU H Uoo = span(v1(oo),---,vm(oo)).

Proposition 6.3 is proved. 0

As a consequence we have

Corollary 6.4 Let L E A(n) and A E U(n) \ {0}. Then there exists Loo E 1,, such
that

lim e‘rAL = Loo

T—‘OIO

Remark 6.5 In [N2] we showed that the flow L H eAtL on A(n) is the gradient
flow of a Zg-perfect Morse function. Moreover the unstable manifolds correspond-
ing to the critical submanifolds of this define a Schubert-type decomposition of the

lagrangian grassmanian.

We now return to our original problem. Thus M is a manifold with boundary
and D is a neck-compatible Dirac operator (throughout this section all Dirac op-
erators on manifolds with boundary will be assummed neck—compatible) and set
M (r) = M U )3 x [0, r]. D has a natural extension D(r) as a neck-compatible Dirac
on M(r) and denote by A’ the CD space of D(r). We are interested in the adiabatic
limit lim...oo A'. As usual set D0 = DIE. For any real number E we denote by
Hf (resp. Hg , HE , Hg , 716”) the subspace of L2(£o) spanned by eigenvectors
corresponding to eigenvalues > E ( 2 E, < E, S E and resp. in I—IEI, IEI I). In

the sequel we will frequently use the following technical result.

Lemma 6.6 For any U C L2(Eo) finite dimensional subspace and any real E the
pair (A’(D),'Hl>'3 G) U) is Fredholm.

55

For a proof of this lemma we refer to [BW4]. For nonnegative E the space ”Hg"
is an isotropic subspace of L2(80). By the above lemma the pair (A'(D),’H§) is
Fredholm so according to Lemma 3.9 we can construct the symplectic reduction
of A’ mod Hg:
r -E
L; = (A 21:3 l. (6.2)
(The symplectic reduction of A = A0 mod HI); will be denoted by L 3). These are

 

lagrangian subspaces in the symplectic vector space Hg. Set A E = DOIug-

Lemma 6.7 The set
.N'(D) = {E20/A(n)nHE = 0}

is a nonempty, closed, unbounded interval.

Proof Consider an increasing sequence E, H 00. Using Lemma 6.6 we obtain a

decreasing sequence of finite dimensional vector spaces
Un = A (1 Hi".

In particular there exists an m > 0 such that

On the other hand DU" 2 0. Thus Um = 0 and therefore Em E N (D) Since
the spectrum of Do is discrete we deduce that N (D) is closed. It is an unbounded

interval because (HE) Ex) is a decreasing family of (isotropic) subspaces of L2(Eo).

0

Definition 6.8 The set N (D) is called the nonresonance range of D.

u(D) = minN(D) is called the nonresonance level of D. When 1/(D) = 0 i.e.

N (D) = [0, 00) the operator D is called nonresonant.

56

We can now formulate the main result of this chapter which shows that the family

A' has alimitas rH 00.

Theorem 6.9 Let M and D as above and E Z V(D). As r H 00
A” H L? EB ’HEE

where

00 _ - r _ ' -—rAE
E — rllrg LE — Tlggloe LE.

Proof Fix E E M(D). The proof is carried out in several steps.
Step1: A dynamical description of A"

Let 8. be the extension of 8 to M(r) and K(r) = K1/2(D(r)). For each 0 S s S r
let

T, : K(r) H L2(8o)

be the restriction map U H U [2)( {8} whose image lies in A3. The CD space Ar can
be equivalently described as A" = T0(K(r)). By Proposition 4.4, To : K(r) H A’
is bijective with continuous inverse. These traces define a backward translation

operator G. : A' H A0 defined as the composition
a. : A’ Ti: K(r) 33. A0. (6.3)
On the cylindrical portion C. = Z x [0,r) of M(r), D(r) has the form

D(r) = c(ds)(ai:: + D0).

Thus any U E K(r) satisfies on C. an evolution like equation

DU=2U+D0U=0.
Us

57

For any u E L2(80) we write u = u+ + an + u- according to the spectral decom-

position

L280 = ”HE ea H5 69 H25
which is independent of 3. Thus we can decompose U(s) = T,U as
U(S) = U(S)+ + U(8)o + U(3)—-

Each of these three pieces satisfies the same evolution-like equation as U (formally
U(s) : e"D° U(0)). Since the spectrum of D0 is discrete we can find a > 0 such
that the set [—u, —E) U (E, [1] contains no eigenvalues of D0. Then we deduce (by

standard Fourier techniques)
I(T9U)+I S const. exp(-#8)|(T0U)+| (64)
|(T,U)-I 2 const. exp(us)|(ToU)_I. (6.5)
Using (6.4)-(6.5) we deduce that Vu E A'

Iu+I2 S const. exp(—ur)I(G,.u)+I2 (6.6)

Iu_I2 2 const. exp(ur)I(G.u)_I2. (6.7)

Intersecting Ar with the coisotropic subspace HEB we get (by Lemma 6.6) the
finite dimensional space
X’ = A" r) HEE
which leads to the symplectic reduction L'E defined in (6.2). Using the Fourier

decomposition for Do we deduce easily that for any L E R Do restricted to H;

defines a Co- semigroup which we denote by e"D° r 2 0. In particular

Ar = e”D°A

58
y, = e"D°LE = e‘rAELE.

Let L°E° = limrnoo e‘TAELE (which exists by Corollary 6.4) and is a lagrangian in
7157.

Step2: Asymptotic transversality If E 2 u(D) then for r large A' is transverse
to the lagrangian subspace

W = JL°E° + Hi3.

First suppose u. E A' n W. Since J Li? C Hg, u. lies in 71' so its orthogonal
projection a. on Hg; lies in L2; flJL°E°. But L27 converges to L°E° which is transverse
to J§° so E. = 0 for large r. Our nonresonant choice E 2 1/(D) then implies

u, = 0, so for large r

A’er = 0. (6.8)

Now, according to Lemma 6.6 (Ar , W) is a Fredholm pair of lagrangians and so

has index 0 by (2.3). Then (6.8) and the definition of the index imply that A’" and

W span so
A” + W : L2(£0). (6.9)
Step 3
flirgAr = L39. (6.10)
By Step 2 Ar 0 W = 0. Since AT, Lia, L2? have the same dimension we can

represent Ar as the the graph of a bounded linear map
B. :Lg—aw = JL°E° +715

To describe B. we first represent L}; as the graph of a symmetric operator 5', :

L°E° —. Lg? (ee Fig. 6.4)

L1}; = {u+JS,u/uEL°E°}

59

   

 

 

  
 

 

W
fin
JSruI
E ).

Figure 6.4: AT is the graph of B, 2 J5, + h,

where S. H 0 since L' H L°° . Next since A” is clean mod HE there exists a
E E >

bounded linear map h, : L? H Hg such that

T = {u + JSru + h.-(u) / u E L?} , Br(u) = (JSrU,hr(U)).
But recall that

A” = e’”D°A = {e'rAEu + e'”AEJSou + e"D°h0(u) / u E L29}

= {v + Je'AESoe'AEv + e"D°ho(erAEv) / v E L°E°}.

Therefore we have
5, = e'AESoerAE and h,(v) = e‘rD°ho(e'AEv).
Then the estimate
Ilhrll S ||€"D°||u§ llhollllerAEIIHg =3 6"“ e'Ellholl

shows that h, H 0 exponentially (we chose u > E) ; we then deduce (6.10) using

Lemma 6.1.

60

Step 4 Convergence. The conditions (6.8) and (6.9) can be used as in the proof
of Theorem 2.2 to represent A’ as the graph of a bounded selfadjoint operator

M, :L%°$HEE H L‘EEBHZE i.e.
A' = {u-i-JMru/uEL‘EEBHEE}.

M, has a block decomposition

_ S, (—Jh.)* . —E_, -E
M,_(_Jhr C. ),C..’H< 7a.

We already know that S, H 0, h, H 0 and we will now show that ”C,“ H 0. The

theorem will follow from Lemma 6.1.

Remark 6.10 Let Poo denote the orthogonal projection onto L23" which is a closed
Do-invariant subspace of L2(£0). IfU E K(r) then wfs) = FOOT3U satisfies the
o.d.e.

w(s) + AEw(s) = 0 s E [0,r).
In particular if w(r) = POOTTU = 0 then the backward translation w{s) =0 for all
s E [0,r].
For any f E HEE consider

u = U(f) = f+JM.f = f+J(—Jh,.)"f+JC.f e A’.

In particular Poou = 0 and any u E A’ with this property can be written in the

above form. Since J(—Jh.)"‘ : HEE H JD}; we deduce
u(f)_ = f and u(f)+ = JCrf = JC.u_. (6.11)

By Remark 6.10 the backward translation of u - defined in (6.3)— v = G.u E A

satisfies Poov = 0 and as in (6.11) we deduce v+ = JCov_. Co is continuous and

61

we get

Bil S const. (6-12)

lv-l

On the other hand (6.6), (6.7) and (6.11) imply

 

I’U+I I(G,u)+ e'“Iu+I 2 IJCrfI

__ = Z = r11 . 6.13

lv—l I(Gru)—| e-wlu-l Ifl ( l
The relations (6.12) and (6.13) imply that “C.” = O(e‘zr"). Theorem 6.9 is

proved.<>

Theorem 6.9 has many interesting corollaries. We will consider only a special
situation motivated by problems in topology (see [Y]). Assume D is nonresonant
i.e.

l/(D) = 0
In this case we will use the simplified notation
ill-(D) = H2. Hall?) = H8, H+(D) = H:-

Here Ho=ker D is finite dimensional and the spaces Hi are spanned by the pos—
itive/ negative eigenmodes of Do. We call H0 the harmonic space of D. Both Hi
are isotropic subspaces of L2(Eo). The annihilator of Hg, is Ho EB Hi. The corre-

sponding symplectic reduction
L(D) = (A 0 (Ho a Hall /H+ (6.14)

will be called the reduced Cauchy data (RCD) space of D. It can be identified
with a lagrangian in the harmonic space. To see this consider the Atiyah-Patodi-

Singer (APS) boundary value problem i.e.

(D,APS)Z Du=0 inM ROUEH+®H0

62

with adjoint

(D,APS)*: Du =0 e M 12021671,.

One sees that
dim L(D) = ind (D,APS) = 1/2dimHo(D).

This agrees with the APS formula since D is selfadjoint so its index density is 0
and Do has a symmetric spectrum ( it anticommutes with J ) so its eta invariant
vanishes. In [APSl], A(D) 0 (H0 EB H+) was called the space of extended L2
solutions and L(D) was identified with the subspace in Ho of asymptotic values
of extended L2 solutions. Using the reduced CD space L(D) we can form the

asymptotic CD space

A°°(D) = L(D) a ’H_(D).

The definition of the asymptotic CD space is orientation sensitive. Changing the
orientation of M without changing that of 2 will have the effect of replacing H-
with H+ in the above definition. We see that D is nonresonant iff (D, APS)‘ has
only the trivial solution. The pleasant thing in the nonresonance case is that the

finite dimensional dynamics is not present since A5 is identically 0 when E = 0 so

that L(D) ;- L’( D) , Vr 2 0. We deduce immediately the following

Corollary 6.11 Assume that D is nonresonant. Then

lim A” = A°°.

rHoo

Corollary 6.12 Let {D(t) ; 0 S t S 1} be a continuous family of neck-compatible

Dirac operators on M such that each D(t) is nonresonant. Let D’(t) denote their

63

extensions to M(r) and A”(t) denote their CD spaces. 1f dim Ker(D0(t)) is inde-

pendent of t then

lim Ar(t) = A°°(t) uniformly in t.

T—*00

In particular (A°°(t)) is a continuous family of lagrangians in L2(€o)

One can use the existence of an adiabatic limit when computing the spectral
flow. We analyze what happens to the terms in Theorem 5.14 as we “stretch the
neck”. Assume we have a path 7 = D(t) E P such that for every t the operators
D1(t) and D2(t) are nonresonant. We can form the adiabatic deformation M(r) of
(M,g) by replacing the neck N E E x (—1,1) by a longer one N, E E X (-r, r). Let
D’(t) be the obvious extension of D(t) to M(r). Denote by A‘J?°(t) the asymptotic

CD space of Dj(t). We have the following result

Corollary 6.13 Let D(t) be a nonresonant path of neck-compatible Dirac opera-

tors such that dimH0(t) is independent of t. Assume
Are) 0 Are) == 0 ,- = 0.1 (615)
Then for r large enough D'(0) and D’"(1) are invertible and

SF(D'UD = B(Ai’°(t)a A‘2’°(t)) (616)

Proof The fact that D'(0) and D”(1) are invertible for large r follows easilyifrom
(6.14) using “adiabatic analysis” as in Thm.6.9. Alternatively we can quote the
results of [CLM2] from which the above conclusion follows trivially. (6.15) follows

from Theorem 5.14 combined with Corollary 6.12. O

64

The nonresonance of the operators D(t) can be translated symplectically by
saying that A1(t) is clean mod H+(D1(t)) and A2(t) is clean mod H-(D2(t)).

Using the invariance of the Maslov index under clean reductions we deduce

Corollary 6.14 Let D(t) as in Corollary 6.13. Then
SF(D'UD = #(L1(t)a 1320))

for r large enough, where L,-(t) = L(D,(t)) is the RCD space of D,-(t).

This last result generalizes a result of [Y]. In that case the Dirac operators arise as

the deformation complexes of the flat connection equation on a homology 3-sphere.

Finally we want to address a natural question. Assume that D(t) is a path
of neck compatible Dirac operators on M1 and suppose that some of them have
positive nonresonace levels. For simplicity suppose V(D(t)) : V0 > 0 for all t
and that the boundary operators D0(t) = D(t) I); are independent of. Then by

Theorem 6.1 we can find lagrangians L°°(t) in HS" such that

lim A’(t) = L°°(t) an? Vt (6.17)

T—+OO

Is the convergence in (6.17) uniform in t ?

We sketch a simple heuristic argument which suggests that the answer one
should expect is in general negative. Let us specialize and assume that the restric-
tion of Do to VI) = H,0 (henceforth denoted by A) has only simple eigenvalues.
In particular A is invertible because it anticommutes with J. Denote by L"(t) the

symplectic reduction of A’(t) mod Hg? We have seen that

7,.(t) = U(t) = e‘A'Lot = e’A'7o(t) Vt.

65

    

gradient

Figure 6.5: A Morse flow on the lagrangian grassmanian

Denote by Lag(Vo) the lagrangian grassmanian associated to the symplectic space
V0. The results of [N2] (see Remark 6.5) show that 6"" is the negative gradient
flow of some function Lag(Vo). Since A has only simple eigenvalues all the critical
points are nondegenerate. The function has a unique critical point P of index 1.
The stable manifold of this point is a codimension 1 submanifold Z of Lag(Vo)
whose closure 7 is the Poincaré dual of the Maslov index (see Fig. 6.5). Now if we
let the path 70 flow along the gradient lines it will “desintegrate”as r H 00 into a
finite set of critical points. Hence the only time 7,(t) can converge uniformly in t
is when 70 lies entirely in the stable manifold of some critical point. Generically
this has to be the region of attraction of the minimum which is the complement
of 7. Via a small perturbation we may assume 70(0), 71(1) lie in this attraction

region and thus we obtain a Maslov index

#(70) = #7007-

66

This number is stable under small perturbations. In particular if u(7o) # 0 then
the endpoints of 70 will flow towards the minimum and some point on this curve
will flow inside Z towards the critical point P and hence we do not have uniform

convergence.

Chapter 7

The Conley-Zehnder index

We will illustrate the general splitting formula on a simple example arising in the

study of periodic trajectories of hamiltonian equations.

We begin by reviewing the Conley-Zehnder index. For details we refer the
reader to [CZ], [R8] or [SZ].

Let E = E, be the standard symplectic space (R2",wo) where

Wo(£L‘,y) : —(Jv$ay) : ($,Jy)

and

J=[I(: ‘0’"I.

Sym(E) will denote the space of symmetric matrices A : E H E. Set
2,, = C1(Sl,Sym(E)).

Associated to any loop A(0) E 2,, is a selfadjoint l-dimensional Dirac operator

0,, :C°°(S‘, E) a C°°(S‘,E) 11(0) H Jig—Z + A(0)u.

Define
E; = {A E 2,, / KerDA = 0}.
67

68
5.3:, is an open subset of 2,. It has countably many paths components. Let

1,, 0
C, = [50 -€InI E Sym(E).

We think of C, as a constant map 51 H Sym(E). There exists an > 0 such that
for all 0 < e S 50 we have C, E 2;.

Following [CZ] we define an injection
11,, : 7rO(Z;,C,) H Z

called the Conley-Zehnder index.

Its construction is carried out in several steps. First, one shows that any
connected component of 2:, contains a constant loop. Next, if 5(0) E S E Z; is

such a loop then 1 is not an eigenvalue of exp(27rJS) so that
U(JS) fl iZ 2 Ill

The eigenvalues of J 8 occur in pairs (A5). We consider only those pairs of purely
imaginary eigenvalues (ADI). If (e,e) are the corresponding eigenvectors then
520(e,e) is purely imaginary. Here 6'20 denotes the complex bilinear extension of

too to C“. Set

U(A) 2 sign 1m 610(25, e)

and

d(A) = o(A)Im A.

The Conley-Zehnder index of S is

22.15) = Z <1a<lll+§l

AEa(JS)niR

69

where [ , I denotes the integer part and in the above sum each eigenvalue is repeated
as many times as its multiplicity. If JS has no purely imaginary eigenvalues than

set 11,,(5) = 0.

Finally one shows that 31, 32 E S ym(E ) fl 2:, belong to the same component of
2; iff 11,,(51) 2 14(5)) so that V, correctly defines an injection 11,, : 7rO(E;, C,) H Z.

If moreover n 2 2 then 11,, is actually a bijection.

The Conley-Zehnder index has an obvious additivity property

Vn1+n2(A1 @ A2) = Vn1(A1) + Vn2(A2) V1416 2;. , 2: 1,2. (7.1)

Example 7.1 Let E = E1 E (R2,wo) and
_ A1 0
S - I o a]
S E 2“; iff A = det S is not a perfect square. IfA S 0 then J5 has only real

eigenvalues so that 111(5) = 0.

IfA > 0 then the eigenvalues of JS' are A, = :ti\/Z with eigenvectors e, =
(1, IPA/Z). We compute
.x/K

aa(e_, a.) = 22——

A2
so that

U(Ai) = :izsign tr(S) , O(Ai) = signtr(S)\/E.

Hence

14(3) = 2Isigntr(S) a] + 1.

Using the additivity property of the index we deduce

Vn(Ce) = 0 V77. _>_1.

70

The Conley-Zehnder index is a Maslov index in disguise. We briefly describe

this point of view following [D2] or [R8].

Let A(0) E 23;. Denote by (I)(0,00) the path of symplectic matrices satisfying

the initial value problem

{ 21% (01 00)
<I>(60, 00)

JA(0)<I>(0,60) 0,00 6 [0,27r]
IE,

(7.2)
Note that a, satisfies the cocycle condition:
a,(0,,93) = o,(0,,02)-<1>,(9,,03) ve1,9,,0a 6 [0,271].
Denote by I‘(0) the graph of own)
I‘(0) = {(a,<1>(9,0)a) / x 6 Ba}.

P(O) is a lagrangian subspace in E, 6 En endowed with the symplectic structure

—(wo) 6 too. In [D2], [RS] it is proved
”rd/4(9)) = ,1(A, I1(9)) (73)

where A is the diagonal A = {(x,x) / x E En} C En6En. Note that A = F(0) so
the endpoints of this pair of paths are not transversal. The Maslov index can still
be defined in this situation and the Maslov index has all the wished for properties:

path additivity and homotopy (rel endpoints) invariance.

Now consider a path t H A,(9) E 23,, such that Aj(6) E E; forj = 0,1. We get
a path of selfadjoint Dirac operators D(t) = D A,. We want to apply the splitting

formula to this path.

To describe the various CD spaces, consider for each t E [0,1] the path of
symplectic matrices <I>,(0,00) defined as in (7.2) with A = A,. The graphs of
<I>,(0,0) will be denoted by F,(0). Represent S1 as in Fig.7.1.

71

21t=0
Figure 7.1: Splitting S1

The objects defined over the LHS of Fig.7.1 will have “ - ”subscripts and those
on the RHS will have “+” subscripts. Then 8 = S1 x E ,80 = E0 6 E,. The

symplectic form on E0 6 E, will be w = (—w0) 6 we. The CD spaces are
A-(t) = {(<I>,(27r,7r)v,v) / v E En}

A+(t) = {(u,<l>,(7r,0)u) / u E E0}.

Thus A-(t) is the graph of @{1(27r,7r) : E2, 2 E0 H E, and A+(t) is the graph
of (I)(n,0) : E0 H E,. Note that

KerD(t) ¢ 0 4:» A_(t)flA+(t) aé 0
so that the pair (A-(t), A+(t)) has transversal endpoints. For 3 e [7r,27r] set
A:(t) = {(<I>,(27r,s)v,v) / v E E,} , A1(t) = {(u,<I>,(s,O)u) / u E E0}.
These paths define a homotopy
(A-(t)aA+(t)) ~ (Ai(t)aAi(t)) ~ (Asa)

where E, C E0 6 E, is the graph of (I),(27r,0). Using the cocycle condition satis-

fied by (I), we see that the endpoints of (A1(), A1()) stay transversal during the

 

Figure 7.2: Intersecting the resonance divisor

deformation. Hence by the splitting formula we have
SF(D(t)) = utMtllAam) = nasal. (7.4)
We get a loop
7 = {1‘0(0); 96 [0,27r1} + {Eti t6 [0,1]} - {1109); 9 610,276}

in the lagrangian grassmanian of E0 6 E,, as in Fig.7.2, which is clearly a con-
tractible loop. In Fig.7.2 Z; is the resonance divisor determined by A consisting

of all lagrangian subspaces of —E 6 E which intersect A nontrivially. Hence

0 = AM) = t(Aarolo + mast) - maria». (7.5)

Using (7.3) and (7.4) in the above equality we deduce
SF(D(t» = ”rd/11(9)) - ”ad/10(9)) (7-6)

which is precisely the content of Theorem 4.1 of [SZ].

73

We conclude this section with a numerical verification of (7.6). Consider a smooth

path of diagonal matrices

A1(t) 0
tHA,=[ 0 Mill tEI—5,5].

Now form the path of 1-dimensional Dirac operators:

D(t) : C°°(SI;E) H C°°(SI;E) u H Jd—u + A,u(0).

d0
We will discuss two cases. Set A(t) = det A,.
A: The Even Case. We will assume that
A(t) > 0 VItI S e (7.7)
A(0) 79 0 (7.8)
6, = {/A(t) E Z+ 4:) t= 0. (7.9)

We have the following result

Proposition 7.2 D(t) is invertible for allt 75 0 and

SF(D(t); ItI S e) = 2sign(A(0)trA0)

Proof If u E KerD(t) then
(7.10)

Set
B, = JA, = I 0 ”MI.

Then, if u satisfies (7.10) we have

u(0) = eB‘“u(0) , u(0) = e2IB‘.

74

Thus KerD(t) ;£ 0 iff 1 is an eigenvalue of of e2”B‘. Note that 82 = —A1 (we

omit the subscript t for simplicity). A simple computation shows

e86 = Icos60 + ésin6dB.

We now compute (0 = 2n)
det (I — 82w) = det((1— cos 27r6)1 — ésin 2776) = 43in2 n6.

Thus 1 is an eigenvalue of e2’IB‘ iff 6, E Z+ i.e. by (7.9) t = 0. In this case

e27rBO = I and

cos 60 —42 sin 60
_ _ _ 6
KerD(O) span (u, I As sin 60 I , U2 I cos 60 I}.

Note that u, _L u, in L2(Sl). Clearly D(0) = A0 and a simple computation shows

(floui a “2)L2(Sl) = (douz a u1)L2(SI) = 0

 

 

. A 0
(A0111 , ul)L2(Sl) Z 71' A( )
2
- A 0
(A0112 , U2)L2(51) = 71' A( )
1
from which we deduce that
AA—[gl 0 .
. < = ' 2 . : '
SF(D(t), ItI _ e) s1gn 0 NAP) 231gn (A(0)trA0).

Proposition 7.2 is proved. D

B: The Odd Case. We assume

|A(t)l s 1/2 WI 3 e (7.11)

75

A(t) = 0 <=> t: 0 (7.12)
A(O), tr A0 ,5 0. (7.13)

The spectral flow of the family D(t) is computed as in Proposition 7.2.

Proposition 7.3 D(t) is invertible for allt 74 0 and

SF(D(t)) = sign (A(0)tr A0)

Proof The first part is established as before and we deduce KerD(t) 95 0 iff t:
0. From (7.11) and (7.12) we deduce that only one of the eigenvalues of A0 is zero

and say A2(0) 75 0. We deduce
6271'80 : 1+BO
so that
KerD(O) = span{f : (0,1)}.
We compute easily
(AOfaf)L2(Sl) = 27rA2(0) = 27r—

and Proposition 7.3 is now obvious. D.

Note that when A(t) < 0 the operator D(t) is invertible an thus there is no change

in the spectral flow.

The computations in Propositions 7.2-3 have a nice geometric interpretation. Any
diagonal matrix as above can be viewed as a point in the plane (A1, A2) (see Fig.3).

A path of such matrices is a path in this plane. For each integer n we have a curve

 

 

 

\ .
trace 11ne

 

-3 -2 -1 0

Figure 7.3: An easier way to compute the Conley-Zehnder index

H, = {A1 A2 = n2 , n(/\, + A2) 2 0} labelled by n in Fig.12. These curves form
the resonance locus R: if a path crosses one of these curves we get a change
in the spectral flow. The labels in Fig.7.3 also define a transversal orientation of
the resonace locus. If H, , n # 0 we get a 21:2 change in the spectral flow. If a
path crosses H0 away from the origin we get a :tl change. We can partition the
complement of the resonance locus as
R2 \ ’R = U R,
7162
where R0 = {A1 A, < 0} and for n 94 0 R, is the region in between the hyperbolae

H, and 11,-," (here a, = sign n. Define
m : R2 \ 'R H Z

by

0 ,AeRo

m(A) = I aa(2lnI—1) , A61,” ”#0 (7.14)

77

Then the above discussion shows

SF(D(t)) = m(A(1)) — m(A(0)) (7.15)

Comparing (7.14) with the computations in Example 7.1 we see that m(A) =

V1(A) and this is in perfect agreement with (7.6).

Appendix A

The proof of Proposition 5.6

Proposition 5.6 is a consequence of the Sard-Smale theorem. We roughly follow
an outline given by F loer (Prop.3.1 in [F I ) making several necessary modifications
(Floer overlooked the hypothesis in Lemma A.2; fixing this requires applying Sard-
Smale to a modified map). To define it, choose k large enough so that Lfi([0, 1] x
M) H 02([0, 1] x M) (deim M) and set

A = {A E Li(End([0,1] X 6)) /A(t) E Cyl(€) Vt E [0, 1] }
We will parametrize the 2-dimensional planes in L¥(E) by

W = {(6.71) 6 14(5) >< L175) / (6.7%» = 0, l€|L2 = I77|La =1}

This is a Banach manifold. Its tangent space at (5,1)) consists of all pairs <15,

16 E Lf(8) that satisfy

(6.1/2) + (M) = (mt) = (7W) = 0 (A-l)

In the proof of our genericity results we will need the following lemmata. _

Lemma A.1 Let D be a cylindrical Dirac and (f, 7]) E W such that D5 2 D7] = 0.
Then there exists an open subset U C M2 away from the neck such that 6 and 77

are pointwise linearly independent over U.

78

79
Proof By unique continuation the set

s: {xéM/€(x)#0andn(rc)#0}

is open and dense as an intersection of two open and dense sets. Set 52 = S O

(M; \ neck). The set
= {x E 52 / £(x)&n(x) are linearly independent }

is open if nonempty. The Lemma is proved if we show that I 31$ 0. Assume the

contrary. This means there exists a E C°°(Sg) such that
to) = atxlnlw) v x e 82

{,7} 75 0 on 52 so that a 74 0 on 52. Since 6 _L n we deduce from the unique
continuation that a is not constant on 32 i.e. do ,.=é 0 on 52. On the other hand

from Definition 4.1
0 =2 D5 2 Don; 2 D7] + [D,a]n = c(da)77.

This is a contradiction since the Clifford multiplication C(da) is an isomorphism

when do aé 0. Lemma A.l is proved. O

For k 2 0 let 5;, denote the linear space of real , symmetric k x ls: matrices

(s0 2 0)

Lemma A.2 Let 5,77 E R" (k 2 2) two linearly independent vectors. Then for

any vectors u,v E R" satisfying

(6.12) = (nan)

there exists A E S), such that (A5, An) = (u,v).

80

Proof Define
H5," : S), H R2” A H (A£,An)

We have to prove that
RangeHM = V,,, = {(u,v) e R“ x R" / ({,v) = (77,u)}.
Note that V,,,, = (span(-n,{))"' and for any A e S,
((AéaAu).(-m€)) = -(A€a7) +<Amél = 0

so that

RangeHm C V5,, (A.2)

On the other hand
dim RangeHm = dim S), — dimKengm
Since 5 and 77 are linearly independent we can identify
Keer E Sk-2
Thus
dim RangeHm = k(k +1)/2 — (k — 2)(k —1)/2 = 2k —1 = dim ng (A.3)

Lemma A.2 follows from( A.2)and ( A.3). 0

Proof of proposition 5.6 We will apply the Sard-Smale theorem to the smooth
function

F:X = Ax(0,1)xWxRHY = L2(£)XL2(E)
defined by

(A(°)7t1€1771’\)H(D(t)€— A77.00)?) + M)

81
Let Z = F‘1(0). The proof of Proposition 5.6 is done in two steps.
Step 1 Z is a smooth Banach manifold.

To prove this we will use the implicit function theorem. Given 2 E Z we will show
that DF(z) : TzX H Y is onto. More precisely we will show that DF(z) has closed
range and its cokernel is zero. Let 2 = (A,t,.£,77,)\) E Z. Note that this implies
A = 0. Indeed we have D(t)£ = An and D(t)n = —/\6 so that

D(t)??? = —AD(t)t = 4272.

Since D(t) is selfadjoint we deduce ID(t)77I2 = —).2I77I2 which is possible iff /\ = 0.

Now consider the variation on the direction (a,r, (25,111,;1) E T,X.The partial

derivatives of F are

DaF(2)(a) = (d(t)€aa(t)n) (A-4)
DtF(2)(T) = T(/1(t)€./i(t)n) (A-5)
Dtt,t)F(z)(¢.2/2) = (D(t)¢.D(t)z/2) (A.6)
DAF(z)(/1) = u(-7a€) (A-7)

where (b and 7,6 satisfy( A.1).

Since the operator D(t) is elliptic we deduce the range of DF(z) is closed.
Let(u,v) E CokerDF(zo). From ( A.4) and ( A.6) we deduce

(a(t)£,u) + (a(t)n,v) = 0 V a E A (A.8)

(190M, U) + (D(twav) = 0 . (A-9)

V d, w satisfying ( A.1) . Let (e,),,€z be the eigenvectors of D(t) corresponding to

the nonzero eigenvalues. If we let (i = e, and w = 0 in ( A.9) we deduce

(emu) = 0 VnEZ

82
so that u E KerD(t). We deduce similarly that v E KerD(t).

From Lemmata A.1 A.2 and ( A.8) we deduce that on an Open set U C M,

away from the neck

(utehvlwll = c(-7(:v).€(:v)) , Vs e U

for some c E R. By unique continuation the above equality holds for all x E M.
Pairing (u,v) with ( A.7) we get that c = 0 i.e. CokerF(z) = 0. Step 1 is

completed.
Step 2 The natural projection 7r : Z H A is Fredholm with index —1.

It is a standard fact that 77 is Fredholm if and only if
G = (77,F):XHAXY

is Fredholm. Moreover nIz and G have the same index. It suffices to study DC at a
point z E Z of our choice. Thus let 20 = (A0, to,£o,no,0) E Z such that Ao(to) is a
positive cylindrical endomorphism. Hence (60, 710) E W and D(to)§o = D(to)no = 0.
The derivatives of G are given by ( A.4)-( A.7) and

DAn(zo)(a) = a , a E A (A.10)

Again the ellipticity of D(to) implies that DG(zo) has closed range.

Let (a, T, d), 77, u) E KerDG(zo). This means a = 0 ,u = 0 , TA0(t0)§o = TA0(t0)no =
D(to)£o = D(to)7)o = 0. In particular since 65 and 7/2 satisfy ( A.1) they lie in a codi-
mension 3 subspace of kerD(to) x kerD(to). Because A0(to) is positive we deduce

Ao(to)€o , Ao(t0)770 ab 0 and therefore

dim KerDG(zo) = 2dim KerD(to) — 3. (A.11)

83

Let (a,u,v) E CokerDG(zo). We deduce from ( A.4) ,( A.6),( A.10) that

(a(t0)§o,u) + (a(to)no,v) + (a,a)A = 0 V a E A (A.12)
T ((A0(to){o,u) + (A0(t0)77090)) = 0 V 7' E R (A.13)
(Danna) + (Datum) = 0 (A.14)

u(<€,v) - (77.21)) = 0 V11 6 R (A-15)

for all 65 and it) satisfying ( A.1). In particular we deduce from ( A.14) and ( A.15)
that u,v E KerD(to) and (u,v) _L (—770,§0). For any u,v E L2(E) define a =

a(u,v) E A by
(a,a)A = —(a(t0)§o,u)L2(g) + (a(t0)no, 'U)L2(g) V a E A. (A.16)

The existence and uniqueness of a such an a is a consequence of Riesz-Frechet

representation theorem. Set
E = {(a(u,v),u,v) / u,v e kerD(t) a (u,v) .1 (—n0,go) }.

We can now describe the cokernel of none) as

CokerDG(zo) = {(a(u,v),u,v) / (u,v) e E , (A0(t0)€o,u) + (fio(to)7]o,v) = 0}.

Since (2106.950, A0(to)no) 1 (—no,go) and Aoaa) is positive we get

dim CokerDG(zo) = dimE —1 = 2dim KerD(to) — 2. (A.17)

Step 2 follows from ( A.11) and ( A.17). Proposition 5.6 is now a consequence of

Sard-Smale theorem. 0

84

Remark A.3 The rank condition on E is always satisfied since from the represen-

tation theory for Clifford algebras the rank ofa real C(M) - module is even.

Remark A.4 Let D be a cylindrical Dirac operator such that dim KerD 2 2. Set
E = KerD x KerD. Let A E Cyl(8) and (5,1)) E Er) W. We get a point C E Z by
C = ((D +(—1/2)A),to =1/2,€,n,0). The equality

indexDG(C) = —1
has the following interesting consequence :

(Air/177) = (0.0) <=> PrOJE(A€)An) = (0.0)-

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