Topology Seminar: Cihan Okay

The classifying space for commutativity, denoted by B_\text{com} G, of a Lie group G is assembled from commuting tuples in G as a subspace of the usual classifying space BG. The resulting space classifies principal G-bundles whose transition functions generate an abelian subgroup of G whenever they are simultaneously defined. The relationship between the homotopy type of G and the space B_\text{com} G is much more interesting, and non-trivial compared to the case of BG. In this talk, I will present a work, joint with Ben Williams, where we study the mod-\ell homotopy type of B_\text{com} G at a prime \ell. The techniques involve a homotopy colimit decomposition over a topological category generalizing the construction of Adem-Gomez and application of results on mapping spaces between classifying spaces of compact Lie groups due to Dwyer-Wilkerson. We show that for a connected compact Lie group the mod-\ell homotopy type of B_\text{com}G depends on the mod-\ell homotopy type of BG.