On spaces of commuting elements in Lie groups
| Autor: | Cohen, Frederick R., Stafa, Mentor |
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| Rok vydání: | 2014 |
| Předmět: | |
| Zdroj: | Math. Proc. Cambridge Philos. Soc. 161(3) (2016), pp. 381-407 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1017/S0305004116000311 |
| Popis: | The main purpose of this paper is to introduce a method to stabilize certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group $G$, namely $Hom(\mathbb Z^n,G)$. We show that this stabilized 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 $Hom(\mathbb Z^n,G)$ 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 stabilized 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 stabilized space also allows the description of the additive reduced homology of the individual spaces $Hom(\mathbb Z^n,G)$, with the order of the Weyl group inverted. Comment: 27 pages, with an appendix by Vic Reiner |
| Databáze: | arXiv |
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