Mathematisches Kolloquium

Zusammenfassung: Question: 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 G and H? For example, the trivial map on the integers Z--->0 arise in this way via the inclusion $Z\subseteq R$ of the integers in the additive group of real numbers. However, the trivial map on a non commutative group G---> 1 cannot arise from such a normal inclusion of topological groups by taking the induced map on path components. In considering this question one builds from the given map f of discrete groups an associated topological space Q:= H//G= H_hG the "homotopy quotient", and attempts to put a group structure on Q. This is not possible in general, for example if Q has non abelian fundamental group. We will discuss similar constructions which are standard in topology and show that a full answer to the above question is given by a simple extra structure on the given map G-->H and lies with a well known notion of crossed module and an observation due to Quillen. ( joint work with Y. Segev.)

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