# zbMATH — the first resource for mathematics

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
Full Text: