| Abstract: |
This is the final paper in a series of three whose objective is to construct a natural
transformation from the surgery exact sequence of Browder, Novikov, Sullivan and
Wall to a long exact sequence of K-theory groups associated to a certain
C*-algebra extension. In this paper we will complete
the construction and it will turn out
that the relationship between signatures and signature operators is fundamental
to this construction. Briefly, to detect whether a homotopy equivalence of manifolds
is a diffeomorphism, we may examine the mapping cylinder and ask whether this
Poincare space (with boundary) is in fact a manifold (with boundary). In turn,
this question may be addressed analytically by asking whether a suitable analytic
signature associated to the Poincare space is actually the analytic index of some
abstract elliptic operator.
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