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Unstable \(K^1\)-group and homotopy type of certain gauge groups. (English) Zbl 1103.55004
From the introduction: Let \(G\) be a compact connected Lie group and let \(P\) be a principal \(G\)-bundle over a connected finite complex \(B\). We denote by \({\mathcal G}(P)\) the group of \(G\)-equivariant selfmaps of \(P\) covering the identity map on \(B\). \({\mathcal G}(P)\) is called the gauge group of \(P\). For fixed \(B\) and \(G\), M. C. Crabb and W. A. Sutherland [Proc. Lond. Math. Soc., III. Ser. 81, 747–768 (2000; Zbl 1024.55005)] showed that as \(P\) ranges over all principal \(G\)-bundles with base space \(B\), the number of homotopy types of \({\mathcal G}(P)\) is finite. Principal SU\((n)\)-bundles over \(S^4\) are classified by their second Chern class. Denote by \(P_{n,k}\) the principal SU\((n)\) bundle over \(S^4\) with \(c_2(P_{n,k})=k\). A. Kono [Proc. R. Soc. Edinb., Sect. A 117, No. 3/4, 295–297 (1991; Zbl 0722.55008)] showed that \({\mathcal G}(P_{2,k})\) is homotopy equivalent to \({\mathcal G}(P_{2,k'})\) if and only if \((12,k)=(12,k')\). Therefore, when \(B\) is \(S^4\) and \(G= \text{SU}(2)\), there are precisely six homotopy types of \({\mathcal G}(P)\). The purpose of this paper is to show the following results: Theorem 1.1. \({\mathcal G}(P_{3,k})\) is homotopy equivalent to \({\mathcal G}(P_{3,k'})\) if and only if \((24,k)\) is equal to \((24,k')\). Theorem 1.2. If \({\mathcal G} (P_{n,k})\) is homotopy equivalent to \({\mathcal G}(P_{n,k'})\), then \((n(n^2-1),k)=(n(n^2-1),k')\).

MSC:
55P15 Classification of homotopy type
55R15 Classification of fiber spaces or bundles in algebraic topology
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57T99 Homology and homotopy of topological groups and related structures
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