Algebra/Topology Seminar

Title: The complex of affinely commutative sets

Given a Lie group G, Adem, Cohen and Torres built up a filtration of the classifying space of G, BG, using the descending central series of finitely generated free groups. The first space in this filtration is the geometric realization of the simplicial space of commuting elements in G, which we’ll denote by B(Z,G) (here Z stands for the integers). Whenever G is abelian, BG=B(Z,G). A natural question is if the converse holds: if the inclusion B(Z,G)->BG is a homotopy equivalence, is G abelian? In joint work with O. Antolín-Camarena, we give a positive answer to this question for compact connected Lie groups and discrete groups. To do this, in the discrete case, we build a simplicial complex with the same homotopy type as E(Z,G), the homotopy fiber of the inclusion B(Z,G)->BG. E(Z,G) can be written as the homotopy colimit of sets G/A, over the poset of abelian subgroups A of G. For compact and connected Lie groups, we give a simplicial model of E(Z,G) and construct a *commutator* map E(Z,G)->B[G,G] which is null-homotopic if and only if G is abelian.