This paper is on the connecting homomorphism in the long exact homotopy sequence of the evaluation fibration ev
p0 :
C(
P, K)
K
→
K, where
C(
P, K)
K
is the gauge group of a continuous principal
K-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 π
2(
C(
P
k
,
K)
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
(
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
Communicated by: C. Schweigert