Xl l
Both theorems 6.2 and 6.10 where stated for families of Dirac operators with
constant symbol. However in Subsection 6.3 we explain how the same proof extends
to the more general case of varying symbols. Similar results were recently proved by
[DZ] in the complex case, using entirely different methods.
The key ingredient which makes this extension possible is the notion of spectral
section defined in [MP], adapted in the obvious manner to a (p,q) setting. Roughly
speaking a spectral section is a continuous family of Atiyah-Patodi-Singer family.
Using the adiabatic analysis of [N2] we show that the restriction to the boundary of
any family of Dirac operators is a family which admits a spectral section. In particular
this provides a short proof of a very general result stating the cobordism invariance
of arbitrary families of Dirac operators. This arbitrariness is two fold: the space of
parameters is any compact CW complex and the index can live in higher A'-theory.
As explained in Subsection 6.3 the CD spaces induce an "excess of symmetry" on the
boundary operators which leads to the vanishing of the index.
We are very pleased of a technical byproduct contained in Appendix B. There
we deal with the continuity of families of Dirac operators with varying boundary
conditions (thus varying domains). This was one delicate point dealt with in the
paper [FOl] when studying paths of ordinary differential operators (what we called
Floer operators). The authors used a conjugation trick which reduces the problem to
families with constant domain. Unfortunately this trick does not generalize well to
partial differential operators.
Although we were interested only in real case (which is richer from a topological
point of view) the methods we develop extend almost verbatim to the complex case
and we considered it was not worth lengthening the presentation by dealing with both
the real and the complex case.
The paper is divided in six sections and we collected the analytical technicalities
in four appendices.
For the readers convenience we present a very short survey of some basic facts
concerning the not so popular but extremely versatile Kp,g theory of Karoubi. In
Section 3 we introduce the notion of (p, q) lagrangians and show how these can be
organized to produce a classifying space of the A'p,g-theory .
The theoretical heart of this work is contained in the sections 4 and 5 which
describe in detail the symplectic skeleton of K theory. We gathered the applications
to index theory in Sec.6.
We want to mention that (most of) the results of this work were announced in
Acknowledgments I want to thank B. Boss and K. Wojciechowski for their
interest in this work. Their results on the boundary value problems for Dirac operators
made me aware of the K-theoretic relevance of the CD spaces.
Also I want to thank W. Zhang for the preprint [DZ]. It was while reading their
work that I got the idea for the new proof of the cobordism invariance of the index
of families described in Proposition 6.7.
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