Homology and K-theory of dynamical systems

Abstract: A theory of homology for étale groupoids was developed by Crainic and Moerdijk based on simplicial structure of nerves of groupoids, as a companion to Haeflier's theory of cohomology for groupoids. We relate this to another (co)homology of groupoids, namely the operator K-groups of the associated convolution algebra, when the base is totally disconnected. Such a connection was conjectured by Matui through his study of Cantor dynamical systems. Our proof is based on the triangulated categorical structure of groupoid equivariant KK-theory, following the categorical approach to the Baum-Connes conjecture by Meyer and Nest. Along the way we uncover the close connection to Putnam's homology theory for hyperbolic dynamical systems (Smale spaces). Based on joint works with Valerio Proietti.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( slides | video )

Noncommutative Geometry in NYC

Series comments: Noncommutative Geometry studies an interplay between spatial forms and algebras with non-commutative multiplication. Our seminar welcomes talks in Number Theory, Geometric Topology and Representation Theory linked to the context of Operator Algebras. All talks are kept at the entry-level accessible to the graduate students and non-experts in the field. To join us click sju.webex.com/meet/nikolaei (5 min in advance) and igor DOT v DOT nikolaev AT gmail DOT com to subscribe/unsubscribe for the mailing list, to propose a talk or to suggest a speaker. Pending speaker's consent, we record and publish all talks at the hyperlink "video" on speaker's profile at the "Past talks" section. The slides can be posted by providing the organizers with a link in the format "myschool.edu/~myfolder/myslides.pdf". The duration of talks is 1 hour plus or minus 10 minutes.