  This paper is on the connecting homomorphism in the long exact homotopy
sequence of the evaluation fibration $ev_{p_0}:C(P,K)^K \to K$, where $C(P,K)^K
\cong Gau(P)$ is the gauge group of a continuous principal $K$-bundle $P$ over
a closed orientable surface or a sphere. We show that in this cases the
connecting homomorphism in the corresponding long exact homotopy sequence is
given in terms of the Samelson product. As applications, we exploit this
correspondence to get an explicit formula for $\pi_2 (Gau(P_k))$, where $P_k$
denotes the principal $S^3$-bundle over $S^4$ of Chern number $k$ and derive
explicit formulae for the rational homotopy groups $\pi_n (Gau(P)) \otimes \Q$.
