Commuting elements and spaces of homomorphisms

This article records basic topological, as well as homological properties of the space of homomorphisms Hom(π,G) where π is a finitely generated discrete group, and G is a Lie group, possibly non-compact. If π is a free abelian group of rank equal to n, then Hom(π, G) is the space of ordered n–tuples of commuting elements in G. If G = SU(2), a complete calculation of the cohomology of these spaces is given for n = 2, 3. An explicit stable splitting of these spaces is also obtained, as a special case of a more general splitting. Alejandro Adem was partially supported by the NSF and NSERC. Frederick R. Cohen was partially supported by the NSF, grant number 0340575.