Colloquium: Tuesday, April 19, 2 pm. Speaker: Emmanuel Farjoun (Hebrew University). Title: “A homotopy path from groups to topological groups”.

Given a map of discrete groups f:G –> H when is this map induced from a normal subgroup inclusion of topological groups X \subseteq Y by taking path components of X and Y?For example, the trivial map on the integers \mathbb{Z} \to 0 arises in this way via the inclusion \mathbb{Z} \subseteq \mathbb{R} of the integers in the additive group of real numbers. However the trivial map on a non commutative group G \to 1 cannot arise from such a normal inclusion of topological groups.

In considering this question one builds from the given map f of discrete groups an associated topological space Q:= H//G, the homotopy quotient, and attempts to put a group structure on Q. This is not possible in general, in particular if Q has non abelian fundamental group.

We will discuss similar constructions and show that the answer to the above question lies with a well known notion of crossed module and an observation due to Quillen.

This is joint work with Segev.