The Samelson product and rational homotopy for gauge groups

This paper is on the connecting homomorphism in the long exact homotopy sequence of the evaluation fibration ev(P0) : C(P, K)(K) -> K, where C(P, K)(K) is the gauge group of a continuous principal K-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 pi(2)(C(P-k, K)(K)), where P-k denotes the principal S-3-bundle over S-4 of Chem number k and derive explicit formulae for the rational homotopy groups pi(n)(C(P, K)(K)) circle times Q.