Doctor of Philosophy
School of Mathematics and Applied Statistics
We present a detailed account of the well-known theory of foliated manifolds, their holonomy groupoids and their characteristic classes using Chern-Weil theory. We give, for the first time, a characteristic map from the cohomology of the Weil algebra of the general linear group into the cohomology of the full holonomy groupoid of the transverse frame bundle of a transversely orientable foliated manifold. From our characteristic map, we derive a codimension 1 Godbillon-Vey cyclic cocycle for the smooth algebra of the transverse frame holonomy groupoid that is the non-etale analogue of the formula given by Connes and Moscovici . Following this, for transversely orientable foliations of arbitrary codimension, we construct unbounded Kasparov modules that are equivariant for actions of the full holonomy groupoid. Finally we show that in codimension 1, one of these Kasparov modules can be used to construct a semifinite spectral triple for the C*-algebra of the transverse frame holonomy groupoid. We prove an index theorem identifying this semifinite spectral triple with our Godbillon-Vey cyclic cocycle, and relate our results to earlier work by Connes. We give in the appendix the required details for equivariant K K-theory for non-Hausdor groupoids, which do not currently exist in the literature.
MacDonald, Lachlan E., Characteristic Classes of Foliated Manifolds in Noncommutative Geometry, Doctor of Philosophy thesis, School of Mathematics and Applied Statistics, University of Wollongong, 2019. https://ro.uow.edu.au/theses1/696
Unless otherwise indicated, the views expressed in this thesis are those of the author and do not necessarily represent the views of the University of Wollongong.