suggests that the family of boundary value problems may have the same index as
the original family (£)') and thus the splitting formula describes the index of the
family (Bl) in terms of boundary data. This is the approach we take in our search
for a general splitting formula: replace the original family with a family of boundary
problems which has the same index and then describe the index of the family of
boundary problems in terms of "interactions along the boundary".
First, we look at Fredholm operators with (Clifford) symmetries (see Prop. 2.9
for their definition). These were shown (cf.[AS], [Ka2]) to form classifying spaces
Fredp,q for all K-groups. Thus to any continuous family y »- Ty Fredp,q there
is an associated index indv,q(Ty) £ Kp,q(Y). Unfortunately the problem of deciding
whether two families have the same index can be very delicate.
The other side of the story is recovered by generalizing the notion of lagrangian
which is the key notion in this paper. A symplectic space can be viewed as a module
over the algebra C1'0 = C and one can define lagrangians in terms of this structure:
any lagrangian subspace defines a Z2-grading of the (l,0)-structure. The generaliza-
tion is now evident. One considers Cp'9-modules and (p, q)-lagrangians (Sec.3) which
can be viewed as defining "super" structures. The space TCp'q of Fredholm pairs of
infinite dimensional (p, q)-lagrangians is a classifying space for the A'p'9-groups of
Karoubi (this is also proved in [KGLZ] in a disguised form). Thus to any continuous
family y »- (A^A^) £ TLVA (y Y-compact CW complex) one can associate an
element ^^(A^A^ ) G Kp,q(Y) called the generalized Maslov index.
The space J-'£p,q is very abstract and a natural question imposes itself: given two
continuous families in TCp'q (parameterized by the same compact CW-complex Y)
decide whether they have the same generalized Maslov index.
The solution to this problem is the key theoretical result of this paper. The
Clifford modules are formally very similar to the usual symplectic spaces. Standard
symplectic operations have natural correspondents to Clifford modules (which we
called generalized symplectic spaces). In particular, the symplectic reduction process
generalizes to arbitrary Clifford modules and more important to infinite dimensions.
The reduction gives a very efficient way of transforming an infinite dimensional prob-
lem to a finite dimensional one. We proved that two continuous families of Fredholm
pairs of lagrangians are homotopic iff we can symplectically reduce them to nomotopic
families of finite dimensional lagrangians (see Thm. 4.14 for details).
We next look at families of Clifford symmetric Fredholm operators (see Sec.5 for
details) and we ask ourselves the same effectivity question: given two continuous
families of Clifford symmetric Fredholm operators decide whether they have the same
index. The answer to this question is the second main theoretical contribution of this
paper. To approach this problem it is more convenient to think of linear operators in
terms of their graphs. Given
T : Dom (T) C H - H
a closed, densely defined, Clifford symmetric Fredholm operator in an Hilbert Cp,(?+1-
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