Localization of grouplike function and section spaces with compact domain

TL;DR

This paper extends localization theory for function and section spaces to compact metric domains with grouplike structures, generalizing previous CW category results and applying to rational homotopy theory.

Contribution

It generalizes existing localization results to compact metric domains with grouplike structures, including applications to gauge groups and fiberwise groups.

Findings

Extended localization for monoids of fiber-homotopy self-equivalences.

Localization theory for groups of sections of fiberwise groups.

Applications to rational homotopy theory.

Abstract

We extend the standard localization theory for function and section spaces due to Hilton-Mislin-Roitberg and Moller outside the CW category to the case of compact metric domain in the presence of a grouplike structure. We study applications in two cases directly generalizing the gauge group of a principal bundle. We prove an identity for the monoid of fibre-homotopy self-equivalences of a Hurewicz fibration -- due to Gottlieb and Booth-Heath-Morgan-Piccinini in the CW category -- in the compact case. This leads to an extended localization result for this monoid. We also obtain an extended localization theory for groups of sections of a fibrewise group. We give two applications in rational homotopy theory.