An algebraic model for finite loop spaces

TL;DR

This paper extends the theory of p-local compact groups to include classifying spaces of finite loop spaces, providing a unified algebraic framework for their p-local homotopy theory and related structures.

Contribution

It generalizes the existing theory to encompass classifying spaces of finite loop spaces within the p-local compact group framework.

Findings

The theory applies to classifying spaces of finite loop spaces.

The main theorem describes the p-completion of total spaces in certain fibrations.

It unifies the study of p-local homotopy types across various classes of spaces.

Abstract

The theory of p-local compact groups, developed in an earlier paper by the same authors, is designed to give a unified framework in which to study the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups, as well as some other families of a similar nature. It also includes, and in many aspects generalizes, the earlier theory of p-local finite groups. In this paper we show that the theory extends to include classifying spaces of finite loop spaces. Our main theorem is in fact more general and states that in a fibration whose base spaces if the classifying space of a finite group, and whose fibre is the classifying space of a p-local compact group, the total space is, up to p-completion the classifying space of a p-local compact group.