The Samelson Product and Rational Homotopy for Gauge Groups

Details The Samelson Product and Rational Homotopy for Gauge Groups The Samelson Product and Rational Homotopy for Gauge Groups

2005-11-16

arxiv.org/abs/math/0511404v2

Abh. Math. Sem. Univ. Hamburg 77 (2007) 219-228

Mathematics

Algebraic Topology

10 pages, 3 figures

Scientific paper

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 $\pi_n (Gau(P)) \otimes \Q$.

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