Homotopy normal maps (Q441068)

The interplay between homotopy theory and group theory is two way. Perhaps the more usual direction is to look for the group-theoretic analogue of some intriguing homotopy-theoretic notion or problem. This paper, on the other hand, looks at the result of making a group-theoretical idea `homotopical'.NEWLINENEWLINEAn inclusion, \(N\hookrightarrow G\), of (topological) groups is the inclusion of a normal subgroup if, and only if, it is the kernel inclusion of some morphism, \(G\to H\). Analogously, here, a loop map, \(\Omega f: \Omega X\to \Omega Y\), between two loop spaces (which are the homotopical analogue of groups) is said to be homotopy-normal if there is a connected space \(W\) with a map \(\pi:Y\to W\), such that NEWLINE\[NEWLINEX\overset{f}{\to}Y\overset{\pi}{\to}WNEWLINE\]NEWLINE is a homotopy fibration sequence.NEWLINENEWLINEThe aim of the paper is to explore thoroughly, and to develop, the notion of homotopy normal map. In particular it is proved that a loop map, \(\Omega f\), as above, is homotopy-normal if, and only if, there is a simplicial loop space, \(\Gamma_{\bullet}\), with \(\Gamma_0\simeq \Omega Y\) (as loop spaces), and such that the canonical homotopy actions of \(\Omega Y\) on \(\Gamma\) and the homotopy analogue of the construction \(\mathrm{Bar}_{\bullet}(\Omega Y,\Omega X)\), are weakly equivalent. Homotopy actions are also studied in some detail and the relation of these notions to ideas from the theory of Segal spaces is discussed.