Homotopy Normal Maps

TL;DR

This paper extends the concept of normal maps from discrete groups to topological groups and loops, characterizing them via simplicial loop space structures and invariance under homotopy monoidal functors, with applications to localizations of fibrations.

Contribution

It introduces a homotopical analogue of normal maps for topological groups and loops, using simplicial structures and homotopy actions, and proves invariance under certain functors.

Findings

Characterization of homotopy normal maps via simplicial loop space structures

Introduction of a homotopy action concept related to $A_{ abla}$ actions

Short proof of a theorem on localizations of principal fibrations

Abstract

Normal maps between discrete groups $N\rightarrow G$ were characterized [FS] as those which induce a compatible topological group structure on the homotopy quotient $EN\times_N G$. Here we deal with topological group (or loop) maps $N\rightarrow G$ being normal in the same sense as above and hence forming a homotopical analogue to the inclusion of a topological normal subgroup in a reasonable way. We characterize these maps by a compatible simplicial loop space structure on $Bar_\bullet(N,G)$, invariant under homotopy monoidal functors, e.g. Localizations and Completions. In the course of characterizing homotopy normality, we define a notion of a "homotopy action" similar to an $A_{\infty}$ action on a space, but phrased in terms of Segal's 'special $\Delta-$spaces' and seem to be of importance on its own right. As an application of the invariance of normal maps, we give a very short proof to a theorem of Dwyer and Farjoun namely that a localization by a suspended map of a principal fibration of connected spaces is again principal.