Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Generalized group characters and complex oriented cohomology theories

Authors: Michael J. Hopkins, Nicholas J. Kuhn and Douglas C. Ravenel
Journal: J. Amer. Math. Soc. 13 (2000), 553-594
MSC (2000): Primary 55N22; Secondary 55N34, 55N91, 55R35, 57R85
Published electronically: April 26, 2000
MathSciNet review: 1758754
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $BG$ be the classifying space of a finite group $G$. Given a multiplicative cohomology theory $E^{*}$, the assignment

\begin{displaymath}G \longmapsto E^{*}(BG) \end{displaymath}

is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories $E^{*}$, using the theory of complex representations of finite groups as a model for what one would like to know.

An analogue of Artin's Theorem is proved for all complex oriented $E^*$: the abelian subgroups of $G$ serve as a detecting family for $E^*(BG)$, modulo torsion dividing the order of $G$.

When $E^*$ is a complete local ring, with residue field of characteristic $p$and associated formal group of height $n$, we construct a character ring of class functions that computes $\frac{1}{p}E^*(BG)$. The domain of the characters is $G_{n,p}$, the set of $n$-tuples of elements in $G$ each of which has order a power of $p$. A formula for induction is also found. The ideas we use are related to the Lubin-Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, $E_n^*$-theory, etc.

The $n$th Morava K-theory Euler characteristic for $BG$ is computed to be the number of $G$-orbits in $G_{n,p}$. For various groups $G$, including all symmetric groups, we prove that $K(n)^*(BG)$ is concentrated in even degrees.

Our results about $E^*(BG)$ extend to theorems about $E^*(EG\times_G X)$, where $X$ is a finite $G$-CW complex.

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

  • [Ada74] J. F. Adams.
    Stable Homotopy and Generalised Homology.
    University of Chicago Press, Chicago, 1974. MR 53:6534
  • [Ada78] J. F. Adams.
    Infinite Loop Spaces.
    Number 90 in Annals of Mathematics Studies. Princeton University Press, Princeton, 1978. MR 80d:55001
  • [Ada82] J. F. Adams.
    Prerequisites (on equivariant stable homotopy) for Carlsson's Lecture.
    Springer Lecture Notes in Math., 1051:483-532, 1982. MR 86f:57037
  • [Ara82] S. Araki.
    Equivariant stable homotopy theory and idempotents of Burnside rings.
    Publ. RIMS, pages 1193-1212, 1982. MR 84g:57035
  • [Atiyah] M. F. Atiyah.
    Characters and cohomology of finite groups.
    Inst. Hautes Études Sci. Publ. Math., 9:23-64, 1961. MR 26:6228
  • [Ati67] M. F. Atiyah.
    Benjamin Press, New York, 1967. MR 36:7130
  • [AS89] M. F. Atiyah and G. Segal.
    On equivariant Euler characteristics.
    J. Geom. Phys., 6:671-677, 1989. MR 92c:19005
  • [BW89] A. Baker and U. Würgler.
    Liftings of formal groups and the Artinian completion of $v_n^{-1}BP$.
    Math. Proc. Camb. Phil. Soc., 106:511-530, 1989. MR 90i:55008
  • [Bak98] A. Baker.
    Hecke algebras acting on elliptic cohomology.
    In Homotopy via algebraic geometry and group representations (Evanston, IL, 1997), A.M.S. Cont. Math. Series 220:17-26, 1998. MR 99h:55004
  • [BT] R. Bott and C. Taubes.
    On the rigidity theorems of Witten.
    J.A.M.S., 2:137-186, 1989. MR 89k:58270
  • [Die72] T. tom Dieck.
    Kobordismentheorie klassifizierender Räume und Transformationsgruppen,
    Math. Zeit., 126:31-39, 1972. MR 45:7744
  • [Die79] T. tom Dieck.
    Transformation Groups and Representation Theory, volume 766 of Lecture Notes in Mathematics.
    Springer-Verlag, New York, 1979. MR 82c:57025
  • [TtD:87] T. tom Dieck.
    Transformation Groups, volume 8 of de Gruyer Studies in Mathematics.
    de Gruyer, New York, 1987. MR 89c:57048
  • [Haz78] M. Hazewinkel.
    Formal Groups and Applications.
    Academic Press, New York, 1978. MR 82a:14020
  • [Hop89] M. J. Hopkins.
    Characters and elliptic cohomology.
    In S. M. Salamon, B. Steer, and W. A. Sutherland, editors, Advances in Homotopy, volume 139 of London Mathematical Society Lecture Note Series, pages 87-104, Cambridge, 1989. Cambridge University Press. MR 91c:55007
  • [HKR92] M. J. Hopkins, N. J. Kuhn, and D. C. Ravenel.
    Morava $K$-theories of classifying spaces and generalized characters for finite groups.
    Algebraic Topology (San Feliu de Guixols, 1990), Springer Lecture Notes in Math., 1509:186-209, 1992. MR 93k:55008
  • [Hun90] J. R. Hunton.
    The Morava $K$-theories of wreath products.
    Math. Proc. Cambridge Phil. Soc., 107:309-318, 1990. MR 91a:55004
  • [Ill74] S. Illman.
    Whitehead torsion and group actions.
    Ann. Acad. Sc. Fennicae, 588, 1974. MR 51:1407
  • [Kri97] I. Kriz.
    Morava $K$-theory of classifying spaces: some calculations.
    Topology, 36:1247-1273, 1997. MR 99a:55016
  • [KL98] I. Kriz and K. P. Lee.
    Odd degree elements in the Morava $K(n)$ cohomology of finite groups.
    Preprint, 1998. To appear in Topology and its applications.
  • [Kuhn1] N. J. Kuhn.
    Morava K-theories of some classifying spaces.
    Transactions of the American Mathematical Society, 304:193-205, 1987. MR 89d:55013
  • [Kuh89] N. J. Kuhn.
    Character rings in algebraic topology.
    In S. M. Salamon, B. Steer, and W. A. Sutherland, editors, Advances in Homotopy, volume 139 of London Mathematical Society Lecture Note Series, pages 111-126, Cambridge, 1989. Cambridge University Press. MR 91g:55023
  • [Lai79] E. Laitinen.
    On the Burnside ring and stable cohomotopy of a finite group.
    Math. Scand., 44:37-72, 1979. MR 80k:55030
  • [Lan71] P. S. Landweber.
    Complex cobordism of classifying spaces.
    Proceedings of the American Mathematical Society, 27:175-179, 1971. MR 42:3782
  • [LRS] P. S. Landweber, D. C. Ravenel, and R. E. Stong.
    Periodic cohomology theories defined by elliptic curves.
    In The Cech centennial (Boston, MA, 1993), A.M.S. Cont. Math. Series 181:317-337, 1995. MR 96i:55009
  • [Lan78] S. Lang.
    Cyclotomic Fields.
    Graduate Texts in Mathematics. Springer-Verlag, New York, 1978. MR 58:5578
  • [LMS86] L. G. Lewis, J. P. May, and M. Steinberger.
    Equivariant Stable Homotopy Theory, volume 1213 of Lecture Notes in Mathematics.
    Springer-Verlag, New York, 1986. MR 88e:55002
  • [LT65] J. Lubin and J. Tate.
    Formal complex multiplication in local fields.
    Annals of Mathematics, 81:380-387, 1965. MR 30:3094
  • [Mor78] J. Morava.
    Completions of complex cobordism.
    In M.G.Barrett and M.E.Mahowald, editors, Geometric Applications of Homotopy Theory II, Springer Lect. Notes Math. 658:349-361, 1978. MR 80i:57026
  • [Mor88] J. Morava.
    Some Weil representations motivated by algebraic topology.
    In P.S. Landweber, editor, Elliptic Curves and Modular Forms in Algebraic Topology, Springer Lect. Notes Math. 1326:94-106, 1988. MR 91a:57021
  • [Qui71] D. G. Quillen.
    The spectrum of an equivariant cohomology ring, I and II.
    Annals of Mathematics, 94:549-602, 1971. MR 45:7743
  • [Rav82] D. C. Ravenel.
    Morava K-theories and finite groups.
    In S. Gitler, editor, Symposium on Algebraic Topology in Honor of José Adem, Contemporary Mathematics, pages 289-292, Providence, Rhode Island, 1982. American Mathematical Society. MR 83m:55009
  • [Rav86] D. C. Ravenel.
    Complex Cobordism and Stable Homotopy Groups of Spheres.
    Academic Press, New York, 1986. MR 87j:55003
  • [RW80] D. C. Ravenel and W. S. Wilson.
    The Morava $K$-theories of Eilenberg-MacLane spaces and the Conner-Floyd conjecture.
    American Journal of Mathematics, 102:691-748, 1980. MR 81i:55005
  • [Seg68:1] G. Segal.
    Classifying spaces and spectral sequences.
    Inst. Hautes Études Sci. Publ. Math., 34:105-112, 1968. MR 38:718
  • [Seg68:2] G. Segal.
    Equivariant K-theory.
    Inst. Hautes Études Sci. Publ. Math., 34:129-151, 1968. MR 38:2769
  • [Seg71] G. Segal.
    Equivariant stable homotopy theory.
    Actes Congr. Internat. Math., 2:59-63, 1970. MR 54:11319
  • [Serre] J.-P. Serre.
    Représentations Linéaires des Groupes Finis.
    Hermann, Paris, 1967. MR 38:1190
  • [Stre81] C. Stretch.
    Stable cohomotopy and cobordism of abelian groups.
    Math. Proc. Cambridge Phil. Soc., 90:273-278, 1981. MR 82h:55015
  • [Stri98] N. P. Strickland.
    Morava $E$-theory of symmetric groups.
    Topology, 37:757-779, 1998. MR 99e:55008; correction MR 2000a:55010
  • [Tan95] M. Tanabe.
    On Morava $K$-theories of Chevalley groups.
    Amer. J. Math., 117:263-278, 1995. MR 95m:55010
  • [Tau] C. Taubes.
    ${S}\sp{1}$-actions and elliptic genera.
    Comm. Math. Phys., 122:455-526, 1989. MR 90f:58167
  • [TY] M. Tezuka and N. Yagita.
    Cohomology of finite groups and Brown-Peterson cohomology.
    In Algebraic Topology (Arcata, CA, 1986), Springer L.N.Math. 1370:396-408, 1989. MR 90i:55011
  • [Wur86] U. Würgler.
    Commutative ring spectra of characteristic $2$.
    Comm. Math. Helv., 61:33-45, 1986. MR 87i:55008

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 55N22, 55N34, 55N91, 55R35, 57R85

Retrieve articles in all journals with MSC (2000): 55N22, 55N34, 55N91, 55R35, 57R85

Additional Information

Michael J. Hopkins
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Nicholas J. Kuhn
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903

Douglas C. Ravenel
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627

Received by editor(s): July 20, 1999
Received by editor(s) in revised form: January 28, 2000
Published electronically: April 26, 2000
Additional Notes: All three authors were partially supported by the National Science Foundation.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society