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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
##### Keywords:
principal $$G$$-bundle; homotopy equivalent
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