An equivariant smash spectral sequence and an unstable box product
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 Trans. Amer. Math. Soc. 351 (1999), 27632775 Request permission
Abstract:
Let $G$ be a finite group. We construct a first quadrant spectral sequence which converges to the equivariant homotopy groups of the smash product $X \wedge Y$ for suitably connected, based $G$CW complexes $X$ and $Y$. The $E^2$ term is described in terms of a tensor product functor of equivariant $\Pi$algebras. A homotopy version of the nonequivariant Künneth theorem and the equivariant suspension theorem of Lewis are both shown to be special cases of the corner of the spectral sequence. We also give a categorical description of this tensor product functor which is analogous to the description in equivariant stable homotopy theory of the box product of Mackey functors. For this reason, the tensor product functor deserves to be called an “unstable box product”.References

M. André, Méthode simpliciale en algèbre homologique et algèbre commutative, Lecture Notes in Mathematics, 32, Springer, Berlin (1967). MR 0214644
A.K. Bousfield and E.M. Friedlander, Homotopy theory of $\Gamma$spaces, spectra, and bisimplicial sets, Geometric Applications of Homotopy Theory II, Lecture Notes in Mathematics, 658, Springer, Berlin (1978) 80130. MR 0513569
A.K. Bousfield and D.M. Kan, Homotopy Limits, Completions, and Localizations, Lecture Notes in Mathematics, 304, Springer, New York (1972). MR 0365573
G.E. Bredon, Equivariant Cohomology Theories, Lecture Notes in Mathematics, 34, SpringerVerlag, Berlin (1967). MR 0214062
R. Brown and J.L. Loday, van Kampen theorems for diagrams of spaces, with an appendix by M. Zisman, Topology, 26 (1987) 311335. MR 0899052
T. tom Dieck, Transformation Groups, Walter de Gruyter, Berlin, 1987. MR 0889050
W.G. Dwyer, D.M. Kan, and C.R. Stover, An $E^2$ model category structure for pointed simplicial spaces, J. Pure and Applied Algebra, 90 (1993) 137152. MR 1250765
P.J. Huber, Homotopy theory in general categories, Math. Annalen, 144 (1961) 361385. MR 0150184
M. Intermont, An equivariant van Kampen spectral sequence, Topology and its Applications, 79 (1997) 3148. MR 1462605
G.M. Kelly, Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Note Series 64, Cambridge University Press, Cambridge, 1982. MR 0651714
L.G. Lewis, Jr, The Box Product of Mackey Functors, unpublished notes.
L.G. Lewis, Jr, Equivariant EilenbergMacLane spaces and the equivariant Seifertvan Kampen and suspension theorems, Topology and its Applications, 48 (1992) 2561. MR 1195124
L.G. Lewis, Jr, The Equivariant Hurewicz Map, Trans. Amer. Math. Soc., 329 (1992) 433472. MR 1049614
L.G. Lewis, Jr, J.P. May, M. Steinberger, with contributions by J.E. McClure, Equivariant Stable Homotopy Theory, Lecture Notes in Mathematics, 1213, Springer, Berlin (1986). MR 0866482
S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics, 5, Springer, New York, 1971. MR 0354798
J.P. May, Simplicial Objects in Algebraic Topology, Van Nostrand, Princeton, 1967. MR 0222892
G. Segal, Categories and cohomology theories, Topology , 13 (1974) 293312. MR 0353298
G. Segal, Classifying spaces and spectral sequences, Inst. H. Et. Sci. Math., 34 (1968) 105112. MR 0232393
C.R. Stover, A van Kampen spectral sequence for higher homotopy groups, Topology, 29 (1990) 926. MR 1046622
D.G. Quillen, Spectral sequences of a double semisimplicial group, Topology, 5 (1966) 155157. MR 0195097
D.G. Quillen, Homotopical Algebra, Lecture Notes in Mathematics 43, Springer, Berlin (1967). MR 0223432
D.G. Quillen, On the (co)homology of commutative rings, Proc. Symp. Pure Math. XVII, Amer. Math. Soc., Providence, RI (1970) 6587. MR 0257068
G.W. Whitehead, Elements of Homotopy Theory, Graduate Texts in Mathematics, 61, Springer, New York, 1978. MR 0516508
Additional Information
 Michele Intermont
 Affiliation: Department of Mathematics, Kalamazoo College, Kalamazoo, Michigan 49006
 Email: intermon@kzoo.edu
 Received by editor(s): August 18, 1997
 Published electronically: February 15, 1999
 © Copyright 1999 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 351 (1999), 27632775
 MSC (1991): Primary 55P91, 55Q91, 55T99, 18G10; Secondary 55P40, 55Q35, 55U25, 18G15, 55U10
 DOI: https://doi.org/10.1090/S0002994799023764
 MathSciNet review: 1621753