## Equivariant properties of symmetric products

HTML articles powered by AMS MathViewer

- by Stefan Schwede;
- J. Amer. Math. Soc.
**30**(2017), 673-711 - DOI: https://doi.org/10.1090/jams/879
- Published electronically: February 24, 2017
- PDF | Request permission

## Abstract:

The filtration of the infinite symmetric product of spheres by the number of factors provides a sequence of spectra between the sphere spectrum and the integral Eilenberg-Mac Lane spectrum. This filtration has received a lot of attention, and the subquotients are interesting stable homotopy types. While the symmetric product filtration has been a major focus of research since the 1980s, essentially nothing was known when one adds group actions into the picture. We investigate the equivariant stable homotopy types, for compact Lie groups, obtained from this filtration of infinite symmetric products of representation spheres. The situation differs from the non-equivariant case; for example, the subquotients of the filtration are no longer rationally trivial and on the zeroth equivariant homotopy groups an interesting filtration of the augmentation ideals of the Burnside rings arises. Our method is by global homotopy theory; i.e., we study the simultaneous behavior for all compact Lie groups at once.## References

- G. Z. Arone and W. G. Dwyer,
*Partition complexes, Tits buildings and symmetric products*, Proc. London Math. Soc. (3)**82**(2001), no. 1, 229–256. MR**1794263**, DOI 10.1112/S0024611500012715 - J. M. Boardman,
*On Stable Homotopy Theory and Some Applications*. Ph.D. thesis, University of Cambridge (1964). - J. M. Boardman and R. M. Vogt,
*Homotopy-everything $H$-spaces*, Bull. Amer. Math. Soc.**74**(1968), 1117–1122. MR**236922**, DOI 10.1090/S0002-9904-1968-12070-1 - Anna Marie Bohmann,
*Global orthogonal spectra*, Homology Homotopy Appl.**16**(2014), no. 1, 313–332. MR**3217308**, DOI 10.4310/HHA.2014.v16.n1.a17 - Armand Borel,
*Seminar on transformation groups*, Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, NJ, 1960. With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. MR**116341** - P. E. Conner and E. E. Floyd,
*Differentiable periodic maps*, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 33, Springer-Verlag, Berlin-Göttingen-Heidelberg; Academic Press, Inc., Publishers, New York, 1964. MR**176478** - Albrecht Dold and René Thom,
*Quasifaserungen und unendliche symmetrische Produkte*, Ann. of Math. (2)**67**(1958), 239–281 (German). MR**97062**, DOI 10.2307/1970005 - Mark Feshbach,
*The transfer and compact Lie groups*, Trans. Amer. Math. Soc.**251**(1979), 139–169. MR**531973**, DOI 10.1090/S0002-9947-1979-0531973-8 - J. P. C. Greenlees and J. P. May,
*Localization and completion theorems for $M\textrm {U}$-module spectra*, Ann. of Math. (2)**146**(1997), no. 3, 509–544. MR**1491447**, DOI 10.2307/2952455 - Nicholas J. Kuhn,
*A Kahn-Priddy sequence and a conjecture of G. W. Whitehead*, Math. Proc. Cambridge Philos. Soc.**92**(1982), no. 3, 467–483. MR**677471**, DOI 10.1017/S0305004100060175 - Kathryn Lesh,
*A filtration of spectra arising from families of subgroups of symmetric groups*, Trans. Amer. Math. Soc.**352**(2000), no. 7, 3211–3237. MR**1707701**, DOI 10.1090/S0002-9947-00-02610-6 - L. G. Lewis, Jr., J. P. May, and M. Steinberger,
*Equivariant stable homotopy theory.*Lecture Notes in Mathematics, Vol. 1213, Springer-Verlag, 1986. - M. A. Mandell, J. P. May, S. Schwede, and B. Shipley,
*Model categories of diagram spectra*, Proc. London Math. Soc. (3)**82**(2001), no. 2, 441–512. MR**1806878**, DOI 10.1112/S0024611501012692 - M. A. Mandell and J. P. May,
*Equivariant orthogonal spectra and $S$-modules*, Mem. Amer. Math. Soc.**159**(2002), no. 755, x+108. MR**1922205**, DOI 10.1090/memo/0755 - J. Peter May,
*$E_{\infty }$ ring spaces and $E_{\infty }$ ring spectra*, Lecture Notes in Mathematics, Vol. 577, Springer-Verlag, Berlin-New York, 1977. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave. MR**494077** - J. P. May,
*Equivariant homotopy and cohomology theory*, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With contributions by M. Cole, G. Comezaña, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. MR**1413302**, DOI 10.1090/cbms/091 - M. C. McCord,
*Classifying spaces and infinite symmetric products*, Trans. Amer. Math. Soc.**146**(1969), 273–298. MR**251719**, DOI 10.1090/S0002-9947-1969-0251719-4 - Stephen A. Mitchell and Stewart B. Priddy,
*Stable splittings derived from the Steinberg module*, Topology**22**(1983), no. 3, 285–298. MR**710102**, DOI 10.1016/0040-9383(83)90014-9 - Minoru Nakaoka,
*Cohomology mod $p$ of symmetric products of spheres*, J. Inst. Polytech. Osaka City Univ. Ser. A**9**(1958), 1–18. MR**121788** - Goro Nishida,
*The transfer homomorphism in equivariant generalized cohomology theories*, J. Math. Kyoto Univ.**18**(1978), no. 3, 435–451. MR**509493**, DOI 10.1215/kjm/1250522505 - S. Schwede,
*Global homotopy theory.*Book project, available from the author’s homepage. - Graeme Segal,
*The representation ring of a compact Lie group*, Inst. Hautes Études Sci. Publ. Math.**34**(1968), 113–128. MR**248277** - G. B. Segal,
*Equivariant stable homotopy theory*, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars Éditeur, Paris, 1971, pp. 59–63. MR**423340** - Victor Snaith,
*Explicit Brauer induction*, Invent. Math.**94**(1988), no. 3, 455–478. MR**969240**, DOI 10.1007/BF01394272 - Peter Symonds,
*A splitting principle for group representations*, Comment. Math. Helv.**66**(1991), no. 2, 169–184. MR**1107837**, DOI 10.1007/BF02566643 - Tammo tom Dieck,
*Orbittypen und äquivariante Homologie. II*, Arch. Math. (Basel)**26**(1975), no. 6, 650–662. MR**436177**, DOI 10.1007/BF01229795 - Peter Webb,
*Two classifications of simple Mackey functors with applications to group cohomology and the decomposition of classifying spaces*, J. Pure Appl. Algebra**88**(1993), no. 1-3, 265–304. MR**1233331**, DOI 10.1016/0022-4049(93)90030-W

## Bibliographic Information

**Stefan Schwede**- Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
- MR Author ID: 623322
- Email: schwede@math.uni-bonn.de
- Received by editor(s): March 17, 2014
- Received by editor(s) in revised form: May 30, 2016
- Published electronically: February 24, 2017
- © Copyright 2017 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**30**(2017), 673-711 - MSC (2010): Primary 55P91
- DOI: https://doi.org/10.1090/jams/879
- MathSciNet review: 3630085