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Function spaces related to gauge groups. (English) Zbl 0761.55007
Let \(G\) be a topological group with classifying space \(BG\) and let \(P\) be the principal \(G\)-bundle over a CW-complex \(X\) classified by \(f:X\to BG\). The classifying space of the gauge group \(G\) is homotopy equivalent to the component of \(\text{Map}(X,B_ G)\) containing \(f\) [the reviewer, Trans. Am. Math. Soc. 171, 23-50 (1972; Zbl 0251.55018)].
The author gives a method using Samelson products which distinguishes these components. He illustrates the idea when \(X\) is a closed surface and \(G\cong U(n)\). He comes close to classifying the components of \(\text{Map}(X,BU(n))\) up to homotopy. He does this for the completion of the components up to a prime \(p\). The author also considers cases where \(G=SU(n)\). The relationships for homotopy-equivalent components depend only on a simple modulus number theory equation.

MSC:
55P15 Classification of homotopy type
55Q15 Whitehead products and generalizations
55R15 Classification of fiber spaces or bundles in algebraic topology
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