The p-local homotopy types of gauge groups of principal G-bundles over S 4 are classified when G is a compact connected exceptional Lie group without p-torsion in homology except for (G, p) = (E 7 , 5).
IntroductionLet G be a topological group and P be a principal G-bundle over a base X. The gauge group G(P ) is the topological group of G-equivariant self-maps of P covering the identity map of X. As P ranges over all principal G-bundles over X, we get a collection of gauge groups G(P ). In [13], Kono classified the homotopy types in the collection of G(P ) when G = SU(2) and X = S 4 . Since then, the homotopy theory of gauge groups has been deeply developed in connection with mapping spaces and fiberwise homotopy theory incorporating higher homotopy structures. The classification of the homotopy types has been its motivation and considerable progress has been made towards it.Suppose that G is a compact connected simple Lie group. Then there is a one-to-one correspondence between principal G-bundles over S 4 and π 3 (G) ∼ = Z. Let G k denote the gauge group of the principal G-bundle over S 4 corresponding to k ∈ Z. Most of the classification of the homotopy types of gauge groups has been done in this setting, generalizing the situation studied by
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