Infiniteness of \(A_\infty\)-types of gauge groups (Q2797317)

Let \(G\) be a compact connected Lie group. For any principal \(G\)-bundle over \(K\), the gauge group of \(P\) is denoted by \({\mathcal G}(P)\). In [Proc. Lond. Math. Soc. (3) 81, No. 3, 747--768 (2000; Zbl 1024.55005)], \textit{M. C. Crabb} and \textit{W. A. Sutherland} proved that, when \(K\) is a finite complex, then the number of homotopy types of \({\mathcal G}(P)\) as \(H\)-spaces is finite. In [J. Lond. Math. Soc., II. Ser. 85, No. 1, 142--164 (2012; Zbl 1236.55013)], \textit{M. Tsutaya} has generalized this result to the number of homotopy types as \(A_{n}\)-spaces with \(n\) finite.NEWLINENEWLINEIn the paper under review, the authors study the number of homotopy types as \(A_{\infty}\)-spaces. In the case of a compact connected simple Lie group \(G\), they prove: As \(P\) ranges over all principal \(G\)-bundles over \(S^d\), if there are infinitely many isomorphism types for \(P\), then there are also infinitely many number of homotopy types as \(A_{\infty}\)-spaces of gauge groups \({\mathcal G}(P)\).