A classification of the stable type of \(BG\). (English) Zbl 0754.55015
Nishida showed the following theorem: Suppose \(G_ 1\) and \(G_ 2\) are finite groups with Sylow \(p\)-subgroups \(P_ 1\) and \(P_ 2\). Then \(BG_ 1\) stably equivalent at \(p\) to \(BG_ 2\) implies \(P_ 1\cong P_ 2\). The authors prove the converse and much more. They show that \(BG_ 1\) is stably homotopy equivalent to \(BG_ 2\) localized at \(p\) if and only if the representation modules (over \(\text{Out}(Q))\) for every \(p\)-group \(Q\) are isomorphic. This classification gives simple-looking specializations.
Reviewer: D.H.Gottlieb (Lafayette)
MSC:
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
| 20J06 | Cohomology of groups |
| 55P42 | Stable homotopy theory, spectra |
Keywords:
classifying space; \(p\)-localization; outer automorphism; finite groups; Sylow \(p\)-subgroups; stably homotopy equivalent; representation modulesReferences:
| [1] | D. J. Benson and M. Feshbach, Stable splittings of classifying spaces of finite groups, Topology 31 (1992), no. 1, 157 – 176. · Zbl 0752.55008 · doi:10.1016/0040-9383(92)90068-S |
| [2] | Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. · Zbl 0075.24305 |
| [3] | John C. Harris and Nicholas J. Kuhn, Stable decompositions of classifying spaces of finite abelian \?-groups, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 3, 427 – 449. · Zbl 0686.55007 · doi:10.1017/S0305004100065038 |
| [4] | J. Lannes, Sur la cohomologie modulo \? des \?-groupes abéliens élémentaires, Homotopy theory (Durham, 1985) London Math. Soc. Lecture Note Ser., vol. 117, Cambridge Univ. Press, Cambridge, 1987, pp. 97 – 116 (French). |
| [5] | Hans-Werner Henn, Jean Lannes, and Lionel Schwartz, Analytic functors, unstable algebras and cohomology of classifying spaces, Algebraic topology (Evanston, IL, 1988) Contemp. Math., vol. 96, Amer. Math. Soc., Providence, RI, 1989, pp. 197 – 220. · Zbl 0683.55013 · doi:10.1090/conm/096/1022682 |
| [6] | John Martino and Stewart Priddy, The complete stable splitting for the classifying space of a finite group, Topology 31 (1992), no. 1, 143 – 156. · Zbl 0752.55010 · doi:10.1016/0040-9383(92)90067-R |
| [7] | Norihiko Minami, Hecke algebras and cohomotopical Mackey functors, Trans. Amer. Math. Soc. 351 (1999), no. 11, 4481 – 4513. · Zbl 0931.55014 |
| [8] | Goro Nishida, Stable homotopy type of classifying spaces of finite groups, Algebraic and topological theories (Kinosaki, 1984) Kinokuniya, Tokyo, 1986, pp. 391 – 404. · Zbl 0800.55001 |
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