Commuting elements and spaces of homomorphisms (Q2373386)

Let \(\pi\) denote a finitely generated discrete group and \(G\) a finite dimensional Lie group. The paper makes substantial contributions to the understanding of the geometry and cohomology of the spaces \(Hom(\pi,G)\) of group homomorphisms of \(\pi\) into \(G\). If \(\pi\) is a free abelian group of rank equal to \(n\), then \(Hom(\pi,G)\) is the space of ordered \(n\)-tuples of commuting elements in \(G\). In a main theorem (for the orthogonal groups), a decomposition is given of \(Hom(\pi,O(n))\) into non-empty, path-connected, disjoint closed subsets enumerated by the first Stiefel-Whitney class of the homomorphisms, and a lower bound is given for the number of components in the space. For \(G=SU(2)\), a complete calculation of the cohomology of these spaces is given for \(n=2,3\). There are numerous other interesting results in the paper.