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, 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 )??.