Homotopy types of gauge groups related to S³-bundles over S⁴

2017/07/21 by Membrillo-Solis, Ingrid
#54C35 #55P15 (primary) #55R10 #55R25 (secondary) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1707.07022

Abstract

Let M_l,m be the total space of the S³-bundle over S⁴ classified by the element lσ+mρ∈π₄(SO(4)), l,m∈\mathbb Z. In this paper we study the homotopy theory of gauge groups of principal G-bundles over manifolds M_l,m when G is a simply connected simple compact Lie group such that π₆(G)=0. That is, G is one of the following groups: SU(n) (n≥4), Sp(n) (n≥2), Spin(n) (n≥5), F₄, E₆, E₇, E₈. If the integral homology of M_l,m is torsion-free, we describe the homotopy type of the gauge groups over M_l,m as products of recognisable spaces. For any manifold M_l,m with non-torsion-free homology, we give a p-local homotopy decomposition, for a prime p≥ 5, of the loop space of the gauge groups.

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