Crossed modules as maps between connected components of topological groups

2016/04/22 by Emmanuel D. Farjoun, Farjoun, Emmanuel D., Yoav Segev +1
Mathematics · #18G30 #55U10 #57T30 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR) #math.AT #math.GR #msc:18G30 #msc:55U10 #msc:57T30

paper · pdf · doi:10.48550/arxiv.1604.06551

8 pages

arxiv created 2016/04/22 · arxiv updated 2016/04/25

Abstract

The purpose of this note is to observe that a homomorphism of discrete groups f:Γ→ G arises as the induced map π₀(\mathfrakM)→ π₀(\mathfrakX) on path components of some closed normal inclusion of topological groups \mathfrakM⊆ \mathfrakX, if and only if the map f can be equipped with a crossed module structure. In that case an essentially unique realization \mathfrakM⊆ \mathfrakX exists by homotopically discrete topological groups.

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