A note on homotopy types of connected components of Map(S^4,BSU(2))

TL;DR

This paper corrects and completes the proof regarding the homotopy types of connected components of the mapping space from S^4 to BSU(2), and explores divisibility properties related to gauge group classifications.

Contribution

It provides a complete proof of a key lemma and investigates divisibility of certain invariants affecting gauge group classifications.

Findings

Corrected proof of homotopy type classification for Map(S^4,BSU(2))

Established divisibility properties of invariants related to gauge groups

Enhanced understanding of A_i-equivalence types of gauge groups

Abstract

Connected components of $\Map(S^4,B\SU(2))$ are the classifying spaces of gauge groups of principal $\SU(2)$-bundles over $S^4$. Tsukuda [Tsu01] has investigated the homotopy types of connected components of $\Map(S^4,B\SU(2))$. But unfortunately, the proof of Lemma 2.4 in [Tsu01] is not correct for $p=2$. In this paper, we give a complete proof. Moreover, we investigate the further divisibility of $\epsilon_i$ defined in [Tsu01]. In [Tsu], it is shown that divisibility of $\epsilon_i$ have some information about $A_i$-equivalence types of the gauge groups.