On spaces of homomorphisms and spaces of representations

The subject of this talk is the structure of the space of homomorphisms from a free abelian group to a Lie group G as well as quotients spaces given by the associated space of representations.

These spaces of representations admit the structure of a simplicial space at the heart of the work here.

Features of geometric realizations will be developed.

What is the fundamental group or the first homology group of the associated space in case G is a finite, discrete group ?

This deceptively elementary question as well as more global information given in this talk is based on joint work with A. Adem, E. Torres, and J. Gomez.