Subgroups of $p$-divisible groups and centralizers in symmetric groups
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- by Nathaniel Stapleton PDF
- Trans. Amer. Math. Soc. 367 (2015), 3733-3757 Request permission
Abstract:
We give a formula relating the transfer maps for the cohomology theories $E_{n}$ and $C_t$ to the transchromatic generalized character maps of a previous paper by the author. We then apply this to understand the effect of the transchromatic generalized character maps on Strickland’s isomorphism between the Morava $E$-theory of the symmetric group $\Sigma _{p^k}$ (modulo a transfer ideal) and the global sections of the scheme that classifies subgroups of order $p^k$ in the formal group associated to $E_{n}$. This provides an algebro-geometric interpretation to the $C_t$-cohomology of the class of groups arising as centralizers of finite sets of commuting elements in symmetric groups.References
- John Frank Adams, Infinite loop spaces, Annals of Mathematics Studies, No. 90, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978. MR 505692, DOI 10.1515/9781400821259
- Matthew Ando, Isogenies of formal group laws and power operations in the cohomology theories $E_n$, Duke Math. J. 79 (1995), no. 2, 423–485. MR 1344767, DOI 10.1215/S0012-7094-95-07911-3
- M. Behrens and C. Rezk, The Bousfield-Kuhn functor and topological Andre-Quillen cohomology. http://math.mit.edu/ mbehrens/papers/BKTAQ4.pdf.
- Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000), no. 3, 553–594. MR 1758754, DOI 10.1090/S0894-0347-00-00332-5
- S. Marsh, The Morava E-theories of finite general linear groups. arxiv:1001.1949.
- J. P. May, The geometry of iterated loop spaces, Lecture Notes in Mathematics, Vol. 271, Springer-Verlag, Berlin-New York, 1972. MR 0420610, DOI 10.1007/BFb0067491
- Nathaniel J. Stapleton, Transchromatic generalized character maps, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)–University of Illinois at Urbana-Champaign. MR 2996052
- N. J. Stapleton, Transchromatic twisted character maps. submitted.
- Neil P. Strickland, Finite subgroups of formal groups, J. Pure Appl. Algebra 121 (1997), no. 2, 161–208. MR 1473889, DOI 10.1016/S0022-4049(96)00113-2
- N. P. Strickland, Morava $E$-theory of symmetric groups, Topology 37 (1998), no. 4, 757–779. MR 1607736, DOI 10.1016/S0040-9383(97)00054-2
Additional Information
- Nathaniel Stapleton
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- Received by editor(s): June 4, 2013
- Received by editor(s) in revised form: November 8, 2013
- Published electronically: December 10, 2014
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 3733-3757
- MSC (2010): Primary 55N20, 55N22; Secondary 14L05
- DOI: https://doi.org/10.1090/S0002-9947-2014-06344-7
- MathSciNet review: 3314822