This thesis studies the K-theory of groupoid C-algebras and itsapplications to topological dynamics and index theory.
Chapter 1 introduces a homology theory for groupoids admitting an open “computable”subgroupoid. This is part of a work-in-progress project whose objective iscomputing the K-groups of C-algebras associated to hyperbolic dynamics.
Paper A (joint work with Jens Kaad) focuses on the assembly map for principalbundles with fiber a countable discrete group. We derive Atiyah’s L2-index theoremin the general context of flat C-module bundles over compact Hausdorff spaces. Ourapproach does not rely on geometric K-homology but rather on a Chern characterconstruction for Alexander-Spanier cohomology.
Paper B deals with the homology groups for Smale spaces defined by Putnam. Weintroduce a simplicial framework by which the various complexes attached to thistheory can be understood as suitable “symmetric” Moore complexes. We prove theyare all quasi-isomorphic and discuss a parallel with sheaf cohomology by computingthe projective cover of a Smale space.
Appendix A contains an induction-restriction adjunction result in the setting ofequivariant Kasparov categories. As a consequence, the KKG-category is describedthrough a complementary pair of subcategories, and a general formulation of thestrong Baum-Connes conjecture for étale groupoids is given.