Forsyth, Iain Graham
Description
The thesis explores two distinct areas of noncommutative
geometry: factorisation and boundaries. Both of these topics are
concerned with cycles in Kasparov’s KK-theory which are defined
using unbounded operators, and manipulating these cycles. These
unbounded operators generalise the Dirac operators of classical
geometry. The first topic of the thesis is factorisation, which
is a process by which one attempts
to represent the class of an equivariant spectral...[Show more] triple as a
product of two unbounded Kasparov cycles, which, if they exist,
are defined using the group action. We provide sufficient
conditions for factorisation to be achieved for actions by
compact abelian Lie groups. We apply our results to examples from
Dirac operators on manifolds and their noncommutative
theta-deformations. In particular, we show that the equivariant
spectral triple associated to a Dirac operator on the total space
of a compact torus
principal bundle always factorises.
The second topic of the thesis is relative spectral triples,
which can be used to describe (noncommutative) manifolds with
boundary. Whereas spectral triples are defined using self-adjoint
unbounded operators, relative spectral triples are defined using
symmetric unbounded operators. We show that the bounded transform
of a relative spectral triple defines a relative Fredholm module,
and hence a class in relative K-homology.
We use relative spectral triples to investigate the boundary map
in the six-term exact sequence of K-homology. We show that the
boundary of a relative spectral triple has a simple description
in terms of extension theory. With some additional data modelled
on the inward normal of a manifold with boundary, we construct a
triple which is a candidate for a spectral triple representing
the boundary class of a relative spectral triple.
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