UBC Topology Seminar: Bernardo Villarreal Herrera

A classical approach to understanding spaces of homomorphisms is to describe its connected components. We mainly focus in the spaces Hom(N_{n,q},SU(2)), where N_{n,q} denotes the free q-nilpotent group on n-generators. We show that the connected components arising from non-commuting q-nilpotent n-tuples in SU(2) are homeomorphic to RP^3 and we give the exact number of these. We prove it by showing a seemingly unknown result about SU(2) that states: all non-abelian nilpotent subgroups are conjugated to the quaternion group Q_8 or to the generalized quaternion groups Q_{2^q}, of order 2^q. Some applications of this result are the stable homotopy type of Hom(N_{n,q},SU(2)); a homotopy description of the classifying spaces B(q,SU(2)) of transitionally q-nilpotent principal SU(2)-bundles, and its derived versions for SO(3) and U(2). If time permits I'll also show some cohomology calculations for the spaces B(r,Q_{2^q}) for low values of r. This is joint work with Omar Antolín Camarena.