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Tilman Bauer: Homotopy representations of Lie groups, 2-compact groups, and 2-local finite groups

Time: Wed 2020-05-20 13.15 - 14.15

Location: Zoom, meeting ID: 689 4480 8319

Participating: Tilman Bauer, KTH


Let \(G\) be a compact Lie group and \(p\) a prime. A (\(p\)-complete, complex, \(n\)-dimensional) homotopy representation of G is a homotopy class from the classifying space \(BG\) of \(G\) to the \(p\)-completed classigying space \(BU(n)\hat{{}_p}\) of the unitary group \(U(n)\). Every genuine representation \(G \to U(n)\) gives rise to a homotopy representation, but this association is neither surjective nor injective unless \(G\) is an extension of a finite \(p\)-group by a torus (a so-called \(p\)-toral group). In fact, homotopy representations are controlled by representations of such \(p\)-toral subgroups of \(G\). Every compact Lie group has a maximal \(p\)-toral subgroup, which is a generalization of a Sylow subgroup of a finite group, and a representation of this Sylow subgroup extends to one of \(G\) if certain invariance conditions are fulfilled and a lifting problem for diagrams in the homotopy category is solvable.

In my talk, I will show how to construct homotopy representations computationally. Particular focus will be on the Dwyer-Wilkerson \(2\)-compact group \(G_3\) and some subgroups of it. \(G_3\) is not quite a compact Lie group but looks like a \(2\)-completion of one, and the subgroups look like finite simple groups but aren't. The smallest currently known nontrivial homotopy representation of \(G_3\) has dimension \(2^{46}\).

I will not assume any knowledge about \(p\)-compact groups, \(p\)-local finite groups or fusion systems from the audience.