Homotopy normal maps

Prezma, Matan

doi:10.2140/agt.2012.12.1211

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Homotopy normal maps

Matan Prezma

Algebraic & Geometric Topology 12 (2012) 1211–1238

DOI: 10.2140/agt.2012.12.1211

Abstract

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 $N \to G$ being “normal” in that they induce a compatible topological group structure on the homotopy quotient $G ∕ ∕ N : = E N \times_{N} G$ . 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.

Keywords

normal subgroup, Segal space, bar construction, localization, completion, homotopy monoidal functor

Mathematical Subject Classification 2010

Primary: 55P35, 18D10

Secondary: 18G55, 55U10, 55U15, 55U30, 55U35

References

Publication

Received: 31 May 2011

Revised: 17 November 2011

Accepted: 29 January 2012

Published: 5 June 2012

Authors

Matan Prezma
Einstein Institute of Mathematics Hebrew University of Jerusalem Giv’at Ram Jerusalem 91904 Israel

Volume 12, issue 2 (2012)