Topology and Geometric Group Theory Seminar
Let $ G $ be a finite-dimensional affine algebraic group defined over a field $ k $ of characteristic 0. For any (discrete) group $ \pi $, the set of all representations of $ \pi $ in $ G $ has a natural structure of an algebraic variety (more precisely, affine k-scheme) called the representation variety $ Rep_G(\pi) $. If $ X $ is a (based) topological space, the representation variety of its fundamental group $ Rep_G[π_1(X)] $ is an important geometric invariant of $X$ that plays a role in many areas of mathematics. In this talk, I will discuss a natural homological extension of this construction, called representation homology, that takes into account a higher homotopy information on spaces and has good functorial properties. The representation homology turns out to be computable (in terms of known invariants) in a number of interesting cases (including simply-connected spaces, Riemann surfaces, link complements, lens spaces, ...), some of which I will examine in detail. Time permitting, I will also explain the relation of representation homology
to other homology theories associated with spaces, such as higher Hochschild homology, $ S^1$-equivariant homology of free loop spaces and the (stable) homology of automorphism groups of the free groups $ F_n $.