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
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