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Homotopy normal maps

Matan Prezma

Algebraic & Geometric Topology 12 (2012) 1211–1238

A group property made homotopical is a property of the corresponding classifying space. This train of thought can lead to a homotopical definition of normal maps between topological groups (or loop spaces).

In this paper we deal with such maps, called homotopy normal maps, which are topological group maps N G being “normal” in that they induce a compatible topological group structure on the homotopy quotient GN := EN ×NG. We develop the notion of homotopy normality and its basic properties and show it is invariant under homotopy monoidal endofunctors of topological spaces, eg localizations and completions. In the course of characterizing normality, we define a notion of a homotopy action of a loop space on a space phrased in terms of Segal’s 1–fold delooping machine. Homotopy actions are “flexible” in the sense they are invariant under homotopy monoidal functors, but can also rigidify to (strict) group actions.

normal subgroup, Segal space, bar construction, localization, completion, homotopy monoidal functor
Mathematical Subject Classification 2010
Primary: 55P35, 18D10
Secondary: 18G55, 55U10, 55U15, 55U30, 55U35
Received: 31 May 2011
Revised: 17 November 2011
Accepted: 29 January 2012
Published: 5 June 2012
Matan Prezma
Einstein Institute of Mathematics
Hebrew University of Jerusalem
Giv’at Ram
Jerusalem 91904