Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T08:50:17.021Z Has data issue: false hasContentIssue false
  • English
  • Français

The chromatic filtration of the Burnside category

Published online by Cambridge University Press:  01 October 2012

MARKUS SZYMIK
MARKUS SZYMIK*
Affiliation:
Mathematisches Institut, Heinrich-Heine-Universität, 40225 Düsseldorf, Germany. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The Segal map connects the Burnside category of finite groups to the stable homotopy category of their classifying spaces. The chromatic filtrations on the latter can be used to define filtrations on the former. We prove a related conjecture of Ravenel's in some cases, and present counterexamples to the general statement.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 154 , Issue 2 , March 2013 , pp. 287 - 302
Copyright
Copyright © Cambridge Philosophical Society 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Adams, J. F.Graeme Segal's Burnside ring conjecture. Symposium on Algebraic Topology in honor of José Adem (Oaxtepec 1981). Contemp. Math. 12 (Amer. Math. Soc., Providence, R.I., 1982), 9–18.CrossRefGoogle Scholar
[2]Adams, J. F., Gunawardena, J. H. and Miller, H. R.The Segal conjecture for elementary abelian p-groups. Topology 24 (1985), 435–460.CrossRefGoogle Scholar
[3]Adams, J. F.Problem session for homotopy theory. Algebraic topology (Arcata, CA, 1986), Lecture Notes in Math. 1370 (Springer, Berlin, 1989), 437–456.Google Scholar
[4]Atiyah, M. F.Characters and cohomology of finite groups. Inst. Hautes Études Sci. Publ. Math. 9 (1961), 23–64.CrossRefGoogle Scholar
[5]Bouc, S.Burnside rings. Handbook of algebra, Vol. 2 (North-Holland, Amsterdam, 2000), 739–804.Google Scholar
[6]Bousfield, A. K.The localization of spectra with respect to homology. Topology 18 (1979), 257–281.CrossRefGoogle Scholar
[7]Brown, E. H. Jr.Framed manifolds with a fixed point free involution. Michigan Math. J. 23 (1976), 257–260.CrossRefGoogle Scholar
[8]Carlsson, G.Equivariant stable homotopy and Segal's Burnside ring conjecture. Ann. of Math. 120 (1984), 189–224.CrossRefGoogle Scholar
[9]Clough, R. R.The /2 cohomology of a candidate for BIm(J). Illinois J. Math. 14 (1970), 424–433.Google Scholar
[10]Davis, D. M.On the Segal conjecture for /2×2. Proc. Amer. Math. Soc. 83 (1981), 619–622.Google Scholar
[11]Devinatz, E. S. and Hopkins, M. J.Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups. Topology 43 (2004), 1–47.CrossRefGoogle Scholar
[12]Dieck, T. tomTransformation groups and representation theory. Lecture Notes in Mathematics 766 (Springer, Berlin, 1979).CrossRefGoogle Scholar
[13]Dress, A.A characterisation of solvable groups. Math. Z. 110 (1969), 213–217.CrossRefGoogle Scholar
[14]Feshbach, M.Essential maps exist from BU to coker(J). Proc. Amer. Math. Soc. 97 (1986), 539–545.Google Scholar
[15]Fiedorowicz, Z. and Priddy, S.Homology of classical groups over finite fields and their associated infinite loop spaces. Lecture Notes in Math. 674 (Springer, Berlin, 1978).CrossRefGoogle Scholar
[16]Greenlees, J. P. C.K-homology of universal spaces and local cohomology of the representation ring. Topology 32 (1993), 295–308.CrossRefGoogle Scholar
[17]Hopkins, M. J.Characters and elliptic cohomology. Advances in homotopy theory (Cortona 1988) London Math. Soc. Lecture Note Ser. 139 (Cambridge University Press, Cambridge, 1989), 87–104.CrossRefGoogle Scholar
[18]Hopkins, M. J., Kuhn, N. J. and Ravenel, D. C.Morava K-theories of classifying spaces and generalized characters for finite groups. Algebraic topology (San Feliu de Guíxols 1990) Lecture Notes in Math. 1509 (Springer, Berlin, 1992), 186–209.Google Scholar
[19]Hopkins, M. J., Kuhn, N. J. and Ravenel, D. C.Generalized group characters and complex oriented cohomology theories. J. Amer. Math. Soc. 13 (2000), 553–594.CrossRefGoogle Scholar
[20]Kuhn, N. J.The mod p K-theory of classifying spaces of finite groups. J. Pure Appl. Algebra 44 (1987), 269–271.CrossRefGoogle Scholar
[21]Kuhn, N. J.The Morava K-theories of some classifying spaces. Trans. Amer. Math. Soc. 304 (1987), 193–205.Google Scholar
[22]Kuhn, N. J.Character rings in algebraic topology. Advances in homotopy theory (Cortona 1988) London Math. Soc. Lecture Note Ser. 139 (Cambridge University Press, Cambridge, 1989), 111–126.Google Scholar
[23]Kriz, I.Morava K-theory of classifying spaces: some calculations. Topology 36 (1997). 1247–1273.CrossRefGoogle Scholar
[24]Kriz, I. and Lee, K. P.Odd-degree elements in the Morava K(n) cohomology of finite groups. Topology Appl. 103 (2000), 229–241.CrossRefGoogle Scholar
[25]Laitinen, E.On the Burnside ring and stable cohomotopy of a finite group. Math. Scand. 44 (1979), 37–72.CrossRefGoogle Scholar
[26]Lewis, L. G., May, J. P. and McClure, J. E.Classifying G-spaces and the Segal conjecture. Current trends in algebraic topology, Part 2 (London, Ont., 1981) CMS Conf. Proc. 2 (Amer. Math. Soc., Providence, R.I., 1982), 165–179.Google Scholar
[27]Mann, B. M., Miller, E. Y. and Miller, H. R.S 1-equivariant function spaces and characteristic classes. Trans. Amer. Math. Soc. 295 (1986), 233–256.Google Scholar
[28]May, J. P.Stable maps between classifying spaces. Conference on algebraic topology in honor of Peter Hilton (Saint John's, Nfld., 1983) Contemp. Math., 37 (Amer. Math. Soc., Providence, RI, 1985), 121–129.CrossRefGoogle Scholar
[29]May, J. P., Snaith, V. P. and Zelewski, P.A further generalization of the Segal conjecture. Quart. J. Math. Oxford Ser. 40 (1989), 457–473.CrossRefGoogle Scholar
[30]Morava, J.Stable homotopy and local number theory. Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) (Johns Hopkins Univ. Press, Baltimore, MD, 1989), 291–305.Google Scholar
[31]Ravenel, D. C.Morava K-theories and finite groups. Symposium on Algebraic Topology in honor of José Adem (Oaxtepec 1981) Contemp. Math. 12 (Amer. Math. Soc., Providence, 1982), 289–292.CrossRefGoogle Scholar
[32]Ravenel, D. C.Localization with respect to certain periodic homology theories. Amer. J. Math. 106 (1984), 351–414.CrossRefGoogle Scholar
[33]Solomon, L.The Burnside algebra of a finite group. J. Combin. Theory 2 (1967), 603–615.CrossRefGoogle Scholar