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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:

Lecturer: Tilman Bauer, KTH

### Abstract

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.

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