Introduction

Consider a closed, compact oriented Riemann manifold (M,g) which is decom-

posed into two manifolds-with-boundary by an oriented hypersurface X): M = Mi Us

Mi Assume moreover that it is given a continuous family (Dy)y^Y oi Dirac type op-

erators on M. Classically, this family has an index in some if group. The problem

we address in this paper is the following:

Describe the index of the family in terms of its behavior on the two pieces of the

decomposition

i.e. we are looking for a splitting formula for the index of a family. If for example the

operators have some symmetries (e.g. they are skew or selfadjoint) then the index lies

in in a higher A'-group (e.g. if all the operators are selfadjoint the index is in K1(Y)).

Thus it is very important to take their symmetries into account. Also, it makes a

difference whether the operators are complex or real. In this paper we will consider

only real operators since they are homotopically more complicated. However all the

techniques extend to the complex case. The natural context which coherently takes

into account all these aspects is that of Fredholm operators with Clifford symmetries

introduced in [AS] and [Ka2],

In a previous paper [Nl] we dealt with a special case of the above splitting problem.

There we considered a path of selfadjoint Dirac operators (Z)*)tg[o,i] on a bundle S of

Clifford modules with a fixed Clifford structure. To any such operator D there is an

associated pair of Cauchy data spaces (CD spaces for brevity) A; (i = 1,2). These

are closed subspaces in L 2 ( £ | E ) defined roughly as follows:

Aj = Az(£) = {c/|

E ;

U e C°°(M), D%U = 0 on M:} i = 1,2.

It turns out that L2(S |E) has a natural symplectic structure and the CD spaces

form Fredholm pairs (cf. Sec.3) of lagrangian subspaces. The space of Fredholm

pairs of lagrangians classifies K1 and an explicit isomorphism K1(S1) — Z can be

constructed, called the Maslov index. Then one shows that the index of the original

path of Dirac operators (also called the spectral flow) equals the Maslov index of the

associated path of CD paces.

An equivalent way of looking at this result is to consider the family of boundary

value problems

B

t

=

{ D\U = V on M1

where D\ = Dl \MX Since (A^A^) is a Fredholm pair the operator Bl is Fredholm

and because A2 is lagrangian Bl is selfadjoint. Moreover keri?* = (ker/)*) \MX which

Received by the editor July 24, 1995.

IX