Ran Levi: p-Local groups and the loop space homology of a small category

ABSTRACT:

Let \(G\) be a finite group, let \(p\) be a prime dividing the order of \(G\) and let \(k\) be a field of characteristic \(p\). In 2009 Benson showed that the homology of the loop space \(\Omega BG^{\wedge_{p}}\) can be computed purely algebraically in terms of a chain complex satisfying certain properties, that as such is unique up to chain homotopy equivalence. The homotopy theory of spaces of the form \(\Omega BG^{\wedge_{p}}\) for \(G\) a finite or more generally a compact Lie group has been of interest for quite some time. Indeed in the early 2000 in a joint project with Carles Broto and Bob Oliver we defined \(p\)-local finite groups and \(p\)-local compact groups. These are algebraic objects that capture the essence of what it means to be a \(p\)-completed classifying space of a finite or a compact Lie group, using only the \(p\)-local information encoded in the group in question. A natural question arising from Benson’s theorem is whether one can make sense from loop space homology of a \(p\)-local group. A positive answer to this question could potentially have implications both within the theory of \(p\)-local groups and more generally. In a current study, joint with Broto and Oliver we showed that in fact loop space homology as an algebraic object makes sense in a much more general context than that of \(p\)-local groups. In this talk I will give a soft introduction to the theory of \(p\)-local groups. Describe Benson’s construction of algebraic loop space homology of a finite group, and then proceed to explain the current state of knowledge in the subject and some interesting open questions.