Homotopy Normal Maps

Normal maps between discrete groups $N\to G$ 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 $N\to G$ 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 $Ba{r}_{\bullet }(N,G)$, 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 $A_{\mathrm{\infty }}$ action on a space, but phrased in terms of Segal’s ‘special $\mathrm{\Delta }-$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.