Homotopy pullback of $A_n$-spaces and its applications to $A_n$-types of gauge groups

TL;DR

This paper develops the homotopy pullback construction for $A_n$-spaces, explores its universal properties, and applies it to classify $A_n$-types of gauge groups, especially for principal $ ext{SU}(2)$-bundles over $S^4$ localized away from 2.

Contribution

It introduces a new homotopy pullback construction for $A_n$-spaces and applies it to classify gauge group $A_n$-types, including a complete classification for certain bundles.

Findings

Existence of a homotopy pullback with universal property for $A_n$-spaces.

A finite CW complex admits an $A_{p-1}$-form but not an $A_p$-form for each prime $p$.

Complete classification of $A_n$-types of gauge groups of principal $ ext{SU}(2)$-bundles over $S^4$ localized away from 2.

Abstract

We construct the homotopy pullback of $A_n$-spaces and show some universal property of it. As the first application, we review the Zabrodsky's result which states that for each prime $p$, there is a finite CW complex which admits an $A_{p-1}$-form but no $A_p$-form. As the second application, we investigate $A_n$-types of gauge groups. In particular, we give a new result on $A_n$-types of the gauge groups of principal $\mathrm{SU}(2)$-bundles over $S^4$, which is a complete classification when they are localized away from 2.