On spaces of commuting elements in Lie groups

Author:

COHEN FREDERICK R.,STAFA MENTOR

Abstract

AbstractThe main purpose of this paper is to introduce a method to “stabilise” certain spaces of homomorphisms from finitely generated free abelian groups to a Lie groupG, namely Hom(ℤn,G). We show that this stabilised space of homomorphisms decomposes after suspending once with “summands” which can be reassembled, in a sense to be made precise below, into the individual spaces Hom(ℤn,G) after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilised space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilised space also allows the description of the additive reduced homology of the individual spaces Hom(ℤn,G), with the order of the Weyl group inverted.

Publisher

Cambridge University Press (CUP)

Subject

General Mathematics

Cited by 8 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Cohomology of the spaces of commuting elements in Lie groups of rank two;Topology and its Applications;2021-11

2. Spaces of commuting elements in the classical groups;Advances in Mathematics;2021-08

3. Homological Stability for Spaces of Commuting Elements in Lie Groups;International Mathematics Research Notices;2020-05-14

4. Commuting matrices and Atiyah's real K‐theory;Journal of Topology;2019-04-17

5. Poincaré series of character varieties for nilpotent groups;Journal of Group Theory;2018-12-18

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