𝑝-subgroups of compact Lie groups and torsion of infinite height in 𝐻*(𝐵𝐺)

Feshbach, Mark

doi:10.1090/S0002-9947-1980-0561834-8

$p$-subgroups of compact Lie groups and torsion of infinite height in $H^{\ast } (BG)$
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Trans. Amer. Math. Soc. 259 (1980), 227-233 Request permission

Abstract:

The relation between elementary abelian p-subgroups of a connected compact Lie group G and the existence of p-torsion in ${H^ {\ast } }(G)$ has been known for some time [B-S]. In this paper we prove that if G is any compact Lie group then ${H^ {\ast } }(BG)$ contains p-torsion of infinite height iff G contains an elementary abelian p-group not contained in a maximal torus. The hard direction is proven using the double coset theorem for the transfer. A third equivalent condition is also given.

References

Armand Borel, Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, Tohoku Math. J. (2) 13 (1961), 216–240 (French). MR 147579, DOI 10.2748/tmj/1178244298
A. Borel and J.-P. Serre, Sur certains sous-groupes des groupes de Lie compacts, Comment. Math. Helv. 27 (1953), 128–139 (French). MR 54612, DOI 10.1007/BF02564557
Mark Feshbach, The transfer and compact Lie groups, Trans. Amer. Math. Soc. 251 (1979), 139–169. MR 531973, DOI 10.1090/S0002-9947-1979-0531973-8
Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602. MR 298694, DOI 10.2307/1970770
Richard G. Swan, The nontriviality of the restriction map in the cohomology of groups, Proc. Amer. Math. Soc. 11 (1960), 885–887. MR 124050, DOI 10.1090/S0002-9939-1960-0124050-2