The Samelson Product and Rational Homotopy for Gauge Groups

Abstract

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 $π_2 (Gau(P_k))$, 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 $π_n (Gau(P)) \otimes \Q$.