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
DOI:
https://doi.org/10.1090/S0894-0347-00-00332-5
Published electronically:
April 26, 2000
MathSciNet review:
1758754
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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 \[ G \longmapsto E^{*}(BG) \] 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.
- J. F. Adams, Stable homotopy and generalised homology, University of Chicago Press, Chicago, Ill.-London, 1974. Chicago Lectures in Mathematics. MR 0402720
- 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
- J. F. Adams, Prerequisites (on equivariant stable homotopy) for Carlsson’s lecture, Algebraic topology, Aarhus 1982 (Aarhus, 1982) Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp. 483–532. MR 764596, DOI https://doi.org/10.1007/BFb0075584
- Shôrô Araki, Equivariant stable homotopy theory and idempotents of Burnside rings, Publ. Res. Inst. Math. Sci. 18 (1982), no. 3, 1193–1212. MR 688954, DOI https://doi.org/10.2977/prims/1195183305
- M. F. Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. 9 (1961), 23–64. MR 148722
- M. F. Atiyah, $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1967. Lecture notes by D. W. Anderson. MR 0224083
- Michael Atiyah and Graeme Segal, On equivariant Euler characteristics, J. Geom. Phys. 6 (1989), no. 4, 671–677. MR 1076708, DOI https://doi.org/10.1016/0393-0440%2889%2990032-6
- Andrew Baker and Urs Würgler, Liftings of formal groups and the Artinian completion of $v_n^{-1}{\rm BP}$, Math. Proc. Cambridge Philos. Soc. 106 (1989), no. 3, 511–530. MR 1010375, DOI https://doi.org/10.1017/S0305004100068249
- Andrew Baker, Hecke algebras acting on elliptic cohomology, Homotopy theory via algebraic geometry and group representations (Evanston, IL, 1997) Contemp. Math., vol. 220, Amer. Math. Soc., Providence, RI, 1998, pp. 17–26. MR 1642886, DOI https://doi.org/10.1090/conm/220/03091
- Raoul Bott and Clifford Taubes, On the rigidity theorems of Witten, J. Amer. Math. Soc. 2 (1989), no. 1, 137–186. MR 954493, DOI https://doi.org/10.1090/S0894-0347-1989-0954493-5
- Tammo tom Dieck, Kobordismentheorie klassifizierender Räume und Transformationsgruppen, Math. Z. 126 (1972), 31–39 (German). MR 298695, DOI https://doi.org/10.1007/BF01580352
- Tammo tom Dieck, Transformation groups and representation theory, Lecture Notes in Mathematics, vol. 766, Springer, Berlin, 1979. MR 551743
- Tammo tom Dieck, Transformation groups, De Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. MR 889050
- Michiel Hazewinkel, Formal groups and applications, Pure and Applied Mathematics, vol. 78, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506881
- Michael J. Hopkins, Characters and elliptic cohomology, Advances in homotopy theory (Cortona, 1988) London Math. Soc. Lecture Note Ser., vol. 139, Cambridge Univ. Press, Cambridge, 1989, pp. 87–104. MR 1055870, DOI https://doi.org/10.1017/CBO9780511662614.010
- Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel, Morava $K$-theories of classifying spaces and generalized characters for finite groups, Algebraic topology (San Feliu de Guíxols, 1990) Lecture Notes in Math., vol. 1509, Springer, Berlin, 1992, pp. 186–209. MR 1185970, DOI https://doi.org/10.1007/BFb0087510
- John Hunton, The Morava $K$-theories of wreath products, Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 2, 309–318. MR 1027783, DOI https://doi.org/10.1017/S0305004100068572
- N. M. Osadčiĭ, The algebras $L^{2}_{n}(\Gamma )$, and the lattice of closed ideals of these algebras, Ukrain. Mat. Ž. 26 (1974), 669–670, 717 (Russian). MR 0365154
- Igor Kriz, Morava $K$-theory of classifying spaces: some calculations, Topology 36 (1997), no. 6, 1247–1273. MR 1452850, DOI https://doi.org/10.1016/S0040-9383%2896%2900049-3 [KL98]krizlee 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.
- Nicholas J. Kuhn, The Morava $K$-theories of some classifying spaces, Trans. Amer. Math. Soc. 304 (1987), no. 1, 193–205. MR 906812, DOI https://doi.org/10.1090/S0002-9947-1987-0906812-8
- Nicholas J. Kuhn, Character rings in algebraic topology, Advances in homotopy theory (Cortona, 1988) London Math. Soc. Lecture Note Ser., vol. 139, Cambridge Univ. Press, Cambridge, 1989, pp. 111–126. MR 1055872, DOI https://doi.org/10.1017/CBO9780511662614.012
- Erkki Laitinen, On the Burnside ring and stable cohomotopy of a finite group, Math. Scand. 44 (1979), no. 1, 37–72. MR 544579, DOI https://doi.org/10.7146/math.scand.a-11795
- Peter S. Landweber, Complex bordism of classifying spaces, Proc. Amer. Math. Soc. 27 (1971), 175–179. MR 268885, DOI https://doi.org/10.1090/S0002-9939-1971-0268885-1
- Peter S. Landweber, Douglas C. Ravenel, and Robert E. Stong, Periodic cohomology theories defined by elliptic curves, The Čech centennial (Boston, MA, 1993) Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 317–337. MR 1320998, DOI https://doi.org/10.1090/conm/181/02040
- Serge Lang, Cyclotomic fields, Springer-Verlag, New York-Heidelberg, 1978. Graduate Texts in Mathematics, Vol. 59. MR 0485768
- L. G. Lewis Jr., J. P. May, M. Steinberger, and J. E. McClure, Equivariant stable homotopy theory, Lecture Notes in Mathematics, vol. 1213, Springer-Verlag, Berlin, 1986. With contributions by J. E. McClure. MR 866482
- Jonathan Lubin and John Tate, Formal complex multiplication in local fields, Ann. of Math. (2) 81 (1965), 380–387. MR 172878, DOI https://doi.org/10.2307/1970622
- Jack Morava, Completions of complex cobordism, Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II, Lecture Notes in Math., vol. 658, Springer, Berlin, 1978, pp. 349–361. MR 513583
- P. S. Landweber (ed.), Elliptic curves and modular forms in algebraic topology, Lecture Notes in Mathematics, vol. 1326, Springer-Verlag, Berlin, 1988. MR 970278
- Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602. MR 298694, DOI https://doi.org/10.2307/1970770
- 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
- Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR 860042
- 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, DOI https://doi.org/10.2307/2374093
- Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112. MR 232393
- Graeme Segal, Equivariant $K$-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129–151. MR 234452
- G. B. Segal, Equivariant stable homotopy theory, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 59–63. MR 0423340
- Jean-Pierre Serre, Représentations linéaires des groupes finis, Hermann, Paris, 1967 (French). MR 0232867
- C. T. Stretch, Stable cohomotopy and cobordism of abelian groups, Math. Proc. Cambridge Philos. Soc. 90 (1981), no. 2, 273–278. MR 620737, DOI https://doi.org/10.1017/S0305004100058734 [Stri98]strickland N. P. Strickland. Morava $E$–theory of symmetric groups. Topology, 37:757–779, 1998. ; correction
- Michimasa Tanabe, On Morava $K$-theories of Chevalley groups, Amer. J. Math. 117 (1995), no. 1, 263–278. MR 1314467, DOI https://doi.org/10.2307/2375045 [Tau]Taubes C. Taubes. ${S}^{1}$–actions and elliptic genera. Comm. Math. Phys., 122:455–526, 1989.
- M. Tezuka and N. Yagita, Cohomology of finite groups and Brown-Peterson cohomology, Algebraic topology (Arcata, CA, 1986) Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 396–408. MR 1000392, DOI https://doi.org/10.1007/BFb0085243
- Urs Würgler, Commutative ring-spectra of characteristic $2$, Comment. Math. Helv. 61 (1986), no. 1, 33–45. MR 847518, DOI https://doi.org/10.1007/BF02621900
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Additional Information
Michael J. Hopkins
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email:
mjh@math.mit.edu
Nicholas J. Kuhn
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903
Email:
njk4x@virginia.edu
Douglas C. Ravenel
Affiliation:
Department of Mathematics, University of Rochester, Rochester, New York 14627
Email:
drav@math.rochester.edu
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