module //, its graph
TT = {(x,Tx) e H xH; xe D(T)}
is a (p+ 1, g + l)-lagrangian with respect to a natural (p+1 , #-}-restructure in H x H.
Moreover (H x {0}, TT) is a Fredholm pair. Thus the map T : T i— TT embeds Fredp,q
in fCp+1,q+1 and in Theorem 5.5 we prove that T is a homotopy equivalence. One
major advantage is that we included in our consideration unbounded operators as
well and this makes Theorem 5.5 a very versatile result. A crucial step in the proof
is the construction of some Clifford symmetric operators which are "generators" in
A"-theory. We called them Floer operators since it seems it was Floer who for the
first time in [F] emphasized their A"-theoretic relevance in the context of symplectic
homology. Now using the symplectic reduction trick for abstract lagrangians we can
reduce the computation of the index of a family of Fredholm operators to a finite
dimensional problem.
Finally we study boundary problems for Dirac operators on a manifold with
boundary. If D is a Clifford symmetric Dirac operator on a bundle E M with
Clifford symmetries, then L2(S\dM) inherits a natural Clifford module structure and
the CD space A(D) of D is a generalized lagrangian subspace. A boundary condition
for D corresponds to a choice of a closed subspace V C L2(S \BM)- This boundary
condition is elliptic iff (A(Z)),V) is a Fredholm pair. The boundary problem thus
obtained displays Clifford symmetries iff V is a generalized lagrangian. Therefore we
look at a family of boundary value problems
f DyU = W on M
v ~ \ U |
a M
Ly y e Y -
Here the parameter space is a compact CW-complex and for each y G Y, Ly is a
generalized lagrangian in L2(S \SM) -depending continuously upon y- such that the
pair (A(Dy), Ly) is Fredholm. The main application of the previous theoretical con-
siderations is Theorem 6.2. A family of boundary problems as above determines two
elements in Kp,g(Y). The first one is an index indp?g(Dy, Ly) if we think of these prob-
lems as defining Fredholm operators. The second element is a measure of the "inter-
action along the boundary" and is the generalized Maslov index /ip+iw+i(A(Dy), Ly).
Theorem 6.2 states a very natural fact:
indP,q(DyjLy) = fip+iiq+i(A(Dy),Ly).
The logical structure of the proof is simple. One first shows that the general case
is equivalent with a special one, namely when M is in fact a cylinder. The cylinder
case is symplectically reduced to a Floer family for which we have already described
the index. The main reason this approach works is the extreme rigidity of Dirac
operators: they satisfy the unique continuation property. As a corollary we deduce
the splitting formula for arbitrary families of Dirac operators (Thm. 6.10) using the
approach outlined in the beginning.
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