The space of commuting elements in an exceptional Lie group and maps between classifying spaces

2024/09/29 by Takeda, Masahiro
#55R37 #57T10 #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.2409.19500

Abstract

Let π be a discrete group, and let G be a compact connected Lie group. Hom(π,G)₀ denotes the null-component of the space of homomorphisms from π to G, and map_*(Bπ,BG)₀ denotes the null-component of the space of maps from Bπ to BG. Since the classifying space functor is continuous, there is a continuous map Θ\colonHom(π,G)₀\tomap_*(Bπ,BG)₀. Atiyah and Bott studied this map when π is a surface group, and proved surjectivity in rational cohomology. In this paper, we obtain the condition that the map Θ is surjective or not in rational cohomology when π is Z^m for m≥ 3 and G is a compact connected Lie group.

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