Abstract (Levi Lorenzo): K-theory is the generalized cohomology theory that has proven incredibly effective at classifying C*- algebras. In this talk, we discuss its dual homology theory, K-homology. K-homology and K-theory admit a bilinear pairing to the integers, called the index pairing, which allows one to learn about K-theory using K-homology and vice versa. In general, this pairing is very difficult to compute; however, if a C*algebra admits a dense subalgebra on which its K-homology is finitely summable, Connes has equipped us with a more computable formula. Given a smooth manifold, M, of dimension n, the K-homology of is finitely summable on for In this way, the summability also sees the dimension of the manifold. It is natural to ask whether similar results hold in the noncommutative setting. Towards this end, Connes showed that the irrational rotation algebras have uniformly summable K-homology. The irrational rotation algebras are examples of crossed product algebras arising from minimal actions on a topological space. In my work, I study other noncommutative algebras and ask the same question. For example, we make the space a Cantor Set and examine the K-homology of crossed products arising from minimal actions on it. In this talk, we discuss progress in such directions.
Abstract (Maggie Reardon): Matui’s HK-conjecture proposes a relationship between the homology groups of a nice enough groupoid and the -theory of the associated reduced -algebra. The conjecture is not true in general and there are a number of counterexamples. However, the conjecture holds for certain classes of groupoids including AF groupoids.
Putnam introduced a new class of groupoids in the paper titled “Some classifiable groupoid -algebras with prescribed -theory”. These new groupoids are related to AF groupoids and this prompts a natural question: does this new class of groupoids satisfy the HK-conjecture?
Finitely Summable K-homology and the HK-conjecture for certain groupoids constructed by Putnam
Finitely Summable K-homology and the HK-conjecture for certain groupoids constructed by Putnam