The Morava -theories of some classifying spaces

Author:
Nicholas J. Kuhn

Journal:
Trans. Amer. Math. Soc. **304** (1987), 193-205

MSC:
Primary 55N22; Secondary 19L99

DOI:
https://doi.org/10.1090/S0002-9947-1987-0906812-8

MathSciNet review:
906812

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a finite abelian -group with classifying space . We compute, in representation theoretic terms, the Morava -theories of the stable wedge summands of . In particular, we obtain a simple, and purely group theoretic, description of the rank of for any finite group with an abelian -Sylow subgroup. A minimal amount of topology quickly reduces the problem to an algebraic one of analyzing truncated polynomial algebras as modular representations of the semigroup .

**[1]**J. F. Adams, J. H. Gunawardena, and H. Miller,*The Segal conjecture for elementary abelian 𝑝-groups*, Topology**24**(1985), no. 4, 435–460. MR**816524**, https://doi.org/10.1016/0040-9383(85)90014-X**[2]**M. F. Atiyah,*Characters and cohomology of finite groups*, Inst. Hautes Études Sci. Publ. Math.**9**(1961), 23–64. MR**0148722****[3]**D. Carlisle, P. Eccles, S. Hilditch, N. Ray, L. Schwartz, G. Walker, and R. Wood,*Modular representations of 𝐺𝐿(𝑛,𝑝), splitting Σ(𝐶𝑃^{∞}×\cdots×𝐶𝑃^{∞}), and the 𝛽-family as framed hypersurfaces*, Math. Z.**189**(1985), no. 2, 239–261. MR**779220**, https://doi.org/10.1007/BF01175047**[4]**D. J. Glover,*A study of certain modular representations*, J. Algebra**51**(1978), no. 2, 425–475. MR**0476841**, https://doi.org/10.1016/0021-8693(78)90116-3**[5]**J. C. Harris and N. J. Kuhn,*Stable decompositions of classifying spaces of finite abelian*-*groups*, Math. Proc. Cambridge Philos. Soc. (submitted).**[6]**M. J. Hopkins and J. H. Smith,*Nilpotence and stable homotopy theory*. II (in preparation).**[7]**Nicholas J. Kuhn,*The mod 𝑝𝐾-theory of classifying spaces of finite groups*, Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), 1987, pp. 269–271. MR**885110**, https://doi.org/10.1016/0022-4049(87)90030-2**[8]**N. J. Kuhn,*The rigidity of*, Proc. Topology year at Univ. of Washington (to appear).**[9]**Nicholas J. Kuhn and Stewart B. Priddy,*The transfer and Whitehead’s conjecture*, Math. Proc. Cambridge Philos. Soc.**98**(1985), no. 3, 459–480. MR**803606**, https://doi.org/10.1017/S0305004100063672**[10]**Stephen A. Mitchell,*Finite complexes with 𝐴(𝑛)-free cohomology*, Topology**24**(1985), no. 2, 227–246. MR**793186**, https://doi.org/10.1016/0040-9383(85)90057-6**[11]**Daniel G. Quillen,*On the associated graded ring of a group ring*, J. Algebra**10**(1968), 411–418. MR**0231919**, https://doi.org/10.1016/0021-8693(68)90069-0**[12]**Douglas C. Ravenel,*Morava 𝐾-theories and finite groups*, Symposium on Algebraic Topology in honor of José Adem (Oaxtepec, 1981), Contemp. Math., vol. 12, Amer. Math. Soc., Providence, R.I., 1982, pp. 289–292. MR**676336****[13]**Douglas C. Ravenel and W. Stephen Wilson,*The Morava 𝐾-theories of Eilenberg-Mac Lane spaces and the Conner-Floyd conjecture*, Amer. J. Math.**102**(1980), no. 4, 691–748. MR**584466**, https://doi.org/10.2307/2374093**[14]**Peter J. Welcher,*Symmetric fibre spectra and 𝐾(𝑛)-homology acyclicity*, Indiana Univ. Math. J.**30**(1981), no. 6, 801–812. MR**632853**, https://doi.org/10.1512/iumj.1981.30.30060**[15]**N. J. Kuhn,*Morova*-*theories and infinite loop spaces*, Proc. 1986 Arcata Topology Conference (to appear).

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
55N22,
19L99

Retrieve articles in all journals with MSC: 55N22, 19L99

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1987-0906812-8

Article copyright:
© Copyright 1987
American Mathematical Society