Equivariant properties of symmetric products

Schwede, Stefan¹

¹Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany

Journal of the American Mathematical Society, Tome 30 (2017) no. 3, pp. 673-711

Résumé

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.

Détail

DOI : 10.1090/jams/879

Bibliographie

[1] Arone, G. Z., Dwyer, W. G. Partition complexes, Tits buildings and symmetric products Proc. London Math. Soc. (3) 2001 229 256

[2] Boardman, J. M., Vogt, R. M. Homotopy-everything 𝐻-spaces Bull. Amer. Math. Soc. 1968 1117 1122

[3] Bohmann, Anna Marie Global orthogonal spectra Homology Homotopy Appl. 2014 313 332

[4] Borel, Armand Seminar on transformation groups 1960

[5] Conner, P. E., Floyd, E. E. Differentiable periodic maps 1964

[6] Dold, Albrecht, Thom, René Quasifaserungen und unendliche symmetrische Produkte Ann. of Math. (2) 1958 239 281

[7] Feshbach, Mark The transfer and compact Lie groups Trans. Amer. Math. Soc. 1979 139 169

[8] Greenlees, J. P. C., May, J. P. Localization and completion theorems for 𝑀𝑈-module spectra Ann. of Math. (2) 1997 509 544

[9] Kuhn, Nicholas J. A Kahn-Priddy sequence and a conjecture of G. W. Whitehead Math. Proc. Cambridge Philos. Soc. 1982 467 483

[10] Lesh, Kathryn A filtration of spectra arising from families of subgroups of symmetric groups Trans. Amer. Math. Soc. 2000 3211 3237

[11] Mandell, M. A., May, J. P., Schwede, S., Shipley, B. Model categories of diagram spectra Proc. London Math. Soc. (3) 2001 441 512

[12] Mandell, M. A., May, J. P. Equivariant orthogonal spectra and 𝑆-modules Mem. Amer. Math. Soc. 2002

[13] May, J. Peter 𝐸_{∞} ring spaces and 𝐸_{∞} ring spectra 1977 268

[14] May, J. P. Equivariant homotopy and cohomology theory 1996

[15] Mccord, M. C. Classifying spaces and infinite symmetric products Trans. Amer. Math. Soc. 1969 273 298

[16] Mitchell, Stephen A., Priddy, Stewart B. Stable splittings derived from the Steinberg module Topology 1983 285 298

[17] Nakaoka, Minoru Cohomology mod 𝑝 of symmetric products of spheres J. Inst. Polytech. Osaka City Univ. Ser. A 1958 1 18

[18] Nishida, Goro The transfer homomorphism in equivariant generalized cohomology theories J. Math. Kyoto Univ. 1978 435 451

[19] Segal, Graeme The representation ring of a compact Lie group Inst. Hautes Études Sci. Publ. Math. 1968 113 128

[20] Segal, G. B. Equivariant stable homotopy theory 1971 59 63

[21] Snaith, Victor Explicit Brauer induction Invent. Math. 1988 455 478

[22] Symonds, Peter A splitting principle for group representations Comment. Math. Helv. 1991 169 184

[23] Tom Dieck, Tammo Orbittypen und äquivariante Homologie. II Arch. Math. (Basel) 1975 650 662

[24] Webb, Peter Two classifications of simple Mackey functors with applications to group cohomology and the decomposition of classifying spaces J. Pure Appl. Algebra 1993 265 304