Representation homology of topological spaces is a natural homological extension of the representation varieties of fundamental groups. In this talk, we will give an interpretation of representation homology as functor homology and explain its relation to other known homology theories associated with spaces (such as higher Hochschild homology, Pontryagin algebras and S1-equivariant homology of free loop spaces). One of our main technical results is a computation of the representation homology of an arbitrary simply connected space (of finite rational type) in terms of its Quillen and Sullivan models. We will also compute explicitly the representation homology of some interesting non-simply connected spaces, such as Riemann surfaces and link complements in R3. In the case of link complements, we obtain a new homological invariant of links analogous to knot contact homology. Time permitting, we will also discuss some applications to representation theory, including an intriguing relation to the celebrated Strong Macdonald Conjecture of Macdonald, Feigin and Hanlon.

This is joint work with Yuri Berest and Wai-kit Yeung.