Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Equivariant BP-cohomology for finite groups


Author: N. Yagita
Journal: Trans. Amer. Math. Soc. 317 (1990), 485-499
MSC: Primary 55N91; Secondary 55N22
DOI: https://doi.org/10.1090/S0002-9947-1990-1002925-3
MathSciNet review: 1002925
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Brown-Peterson cohomology rings of classifying spaces of finite groups are studied, considering relations to the other generalized cohomology theories. In particular, $ {\operatorname{BP} ^{\ast}}(M)$ are computed for minimal nonabelian $ p$-groups $ M$. As an application, we give a necessary condition for the existence of nonabelian $ p$-subgroups of compact Lie groups.


References [Enhancements On Off] (What's this?)

  • [1] L. Evens, A generalization of the transfer map in the cohomology of groups, Trans. Amer. Math. Soc. 108 (1963), 54-65 MR 0153725 (27:3686)
  • [2] D. Johnson and S. Wilson, $ BP$ operations and Morava's extraordinary $ K$-theories, Math. Z. 144 (1975), 55-75. MR 0377856 (51:14025)
  • [3] P. Landweber Conference, flatness and cobordism of classifying spaces, Proc. Aarhus Summer Institute on Algebraic Topology 1970, pp. 256-269. MR 0271964 (42:6845)
  • [4] -, Homological properties of comodules over $ M{U_{\ast}}(MU)$ and $ {\operatorname{BP} _{\ast}}(\operatorname{BP} )$, Amer. J. Math. 98 (1976), 591-610. MR 0423332 (54:11311)
  • [5] D. Quillen, A topological criterion for $ p$-nilpotency, J. Pure Appl. Algebra 4 (1971), 373-376. MR 0318339 (47:6886)
  • [6] -, The spectrum of an equivariant ring. I, II, Ann. of Math. 94 (1971), 549-572, 573-602. MR 0298694 (45:7743)
  • [7] -, Elementary proofs of some results of cobordism theory using Steenrod operation, Adv. in Math. 7 (1971), 29-56. MR 0290382 (44:7566)
  • [8] D. Ravenel, private communication.
  • [9] L. Redéi, Das schiefe Product in der Gruppentheorie, Comment. Math. Helv. 20 (1947), 225-264. MR 0021933 (9:131a)
  • [10] C. Stretch Stable cohomology and cobordism of abelian groups, Math. Proc. Cambridge Philos. Soc. 90 (1981), 273-278. MR 620737 (82h:55015)
  • [11] M. Tezuka and N. Yagita, Cohomology of finite groups and the Brown-Peterson cohomology, Lecture Notes in Math., vol. 1370, Springer. MR 1000392 (90i:55011)
  • [12] N. Yagita, The exact functor theorem for $ {\operatorname{BP} ^{\ast}}/{I_n}$-theory, Proc. Japan Acad. 52 (1976), 1-3. MR 0394631 (52:15432)
  • [13] -, On relations between Brown-Peterson cohomology and the ordinary $ \bmod p$ cohomology theory, Kodai Math. J. 7 (1984), 273-285. MR 744140 (85g:55007)
  • [14] -, On the dimension of spheres whose product admits a free action by a non abelian $ p$-group, Quart. J. Math. Oxford Ser. 36 (1985), 117-127. MR 780356 (86h:57041)
  • [15] A Borel and J. P. Serre, Sur certain sous-groupes des groupes de Lie compacts, Comment. Math. Helv. 27 (1953), 128-139. MR 0054612 (14:948d)
  • [16] E. Friedlander and G. Mislin, Cohomology of classifying spaces of complex Lie groups and related discrete groups, Comment. Math. Helv. 59 (1984), 347-361. MR 761803 (86j:55011)
  • [17] S. Kleinerman, The cohomology of Chevally groups of exceptional Lie type, Mem. Amer. Math. Soc. 268 (1982). MR 668808 (84i:20048)
  • [18] S. Ihara and T. Yokonuma, On the second cohomology groups (Shur-multipliers) of finite reflection groups, J. Fac. Sci. Univ. Tokyo Sect. 11 (1965), 155-171. MR 0190232 (32:7646a)
  • [19] C. B. Thomas, Characteristic classes and the cohomology of finite group, Cambridge Univ. Press, 1987. MR 878978 (88f:20005)
  • [20] M. Hopkins N. Kuhn, and D. Ravenel, Generalized group characters and complex oriented cohomology theories (to appear). MR 1758754 (2001k:55015)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55N91, 55N22

Retrieve articles in all journals with MSC: 55N91, 55N22


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-1002925-3
Keywords: Classifying space $ BG$, cohomology of groups, $ \operatorname{BP} $-theory
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society