The Samelson product and rational homotopy for gauge groups


DOI: 10.1007/BF03173500

Cite this article as:
Wockel, C. Abh. Math. Sem. Univ. Hamburg (2007) 77: 219. doi:10.1007/BF03173500


This paper is on the connecting homomorphism in the long exact homotopy sequence of the evaluation fibration evp0 :C(P, K)KK, 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(Pk,K)K), wherePk denotes the principal S3-bundle over S4 of Chern numberk and derive explicit formulae for the rational homotopy groups πn(C(P,K)K)⊗ℚ.

2000 Mathematics Subject Classification

57T20 57S05 81R10 55P62 

Key words and phrases

bundles over spheres bundles over surfaces gauge groups pointed gauge groups homotopy groups of gauge groups rational homotopy groups of gauge groups evaluation fibration connecting homomorphism Samelson product Whitehead product 

Copyright information

© Mathematische Seminar 2007

Authors and Affiliations

  1. 1.Mathematisches InstitutGeorg-August-Universität GöttingenGöttingenGermany

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