The Samelson product and rational homotopy for gauge groups

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Abstract

This paper is on the connecting homomorphism in the long exact homotopy sequence of the evaluation fibration evp0 :C(P, K) K K, whereC(P, K) K is the gauge group of a continuous principalK-bundle. We show that in the case of a bundle over a sphere or a orientable surface the connecting homomorphism is given in terms of the Samelson product. As applications we get an explicit formula for π2(C(P k ,K) K ), whereP k denotes the principal S3-bundle over S4 of Chern numberk and derive explicit formulae for the rational homotopy groups π n (C(P,K) K )⊗ℚ.

Communicated by: C. Schweigert