Boundaries and equivariant products in unbounded Kasparov theory

Abstract

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 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.