Stably splitting BG

Benson, Dave

doi:10.1090/S0273-0979-96-00656-8

Stably splitting BG
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by Dave Benson PDF

Bull. Amer. Math. Soc. 33 (1996), 189-198 Request permission

Abstract:

In the early nineteen eighties, Gunnar Carlsson proved the Segal conjecture on the stable cohomotopy of the classifying space $BG$ of a finite group $G$. This led to an algebraic description of the ring of stable self-maps of $BG$ as a suitable completion of the “double Burnside ring”. The problem of understanding the primitive idempotent decompositions of the identity in this ring is equivalent to understanding the stable splittings of $BG$ into indecomposable spectra. This paper is a survey of the developments of the last ten to fifteen years in this subject.

References

J. F. Adams, Graeme Segal’s Burnside ring conjecture, Symposium on Algebraic Topology in honor of José Adem (Oaxtepec, 1981), Contemp. Math., vol. 12, Amer. Math. Soc., Providence, R.I., 1982, pp. 9–18. MR 676311
J. F. Adams, J. H. Gunawardena, and H. Miller, The Segal conjecture for elementary abelian $p$-groups, Topology 24 (1985), no. 4, 435–460. MR 816524, DOI 10.1016/0040-9383(85)90014-X
M. F. Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. 9 (1961), 23–64. MR 148722, DOI 10.1007/BF02698718
D. J. Benson and M. Feshbach, Stable splittings of classifying spaces of finite groups, Topology 31 (1992), no. 1, 157–176. MR 1153243, DOI 10.1016/0040-9383(92)90068-S
David Carlisle and Nicholas J. Kuhn, Subalgebras of the Steenrod algebra and the action of matrices on truncated polynomial algebras, J. Algebra 121 (1989), no. 2, 370–387. MR 992772, DOI 10.1016/0021-8693(89)90073-2
Gunnar Carlsson, G. B. Segal’s Burnside ring conjecture for $(\textbf {Z}/2)^{k}$, Topology 22 (1983), no. 1, 83–103. MR 682060, DOI 10.1016/0040-9383(83)90046-0
Gunnar Carlsson, Equivariant stable homotopy and Segal’s Burnside ring conjecture, Ann. of Math. (2) 120 (1984), no. 2, 189–224. MR 763905, DOI 10.2307/2006940
A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. I, Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I., 1961. MR 0132791
Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
Ethan S. Devinatz, Michael J. Hopkins, and Jeffrey H. Smith, Nilpotence and stable homotopy theory. I, Ann. of Math. (2) 128 (1988), no. 2, 207–241. MR 960945, DOI 10.2307/1971440
Jill Dietz, Stable splittings of classifying spaces of metacyclic $p$-groups, $p$ odd, J. Pure Appl. Algebra 90 (1993), no. 2, 115–136. MR 1250764, DOI 10.1016/0022-4049(93)90125-D
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
J. H. C. Gunawardena. Segal’s conjecture for cyclic groups of (odd) prime order. J. T. Knight Prize Essay, Cambridge, 1980.
J. C. Harris. Thesis. University of Chicago, 1985.
John C. Harris and Nicholas J. Kuhn, Stable decompositions of classifying spaces of finite abelian $p$-groups, Math. Proc. Cambridge Philos. Soc. 103 (1988), no. 3, 427–449. MR 932667, DOI 10.1017/S0305004100065038
J. M. Howie, An introduction to semigroup theory, L. M. S. Monographs, No. 7, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. MR 0466355
Jean Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un $p$-groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 135–244 (French). With an appendix by Michel Zisman. MR 1179079, DOI 10.1007/BF02699494
L. G. Lewis, J. P. May, and J. E. McClure, Classifying $G$-spaces and the Segal conjecture, Current trends in algebraic topology, Part 2 (London, Ont., 1981) CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, R.I., 1982, pp. 165–179. MR 686144
Wen Hsiung Lin, On conjectures of Mahowald, Segal and Sullivan, Math. Proc. Cambridge Philos. Soc. 87 (1980), no. 3, 449–458. MR 556925, DOI 10.1017/S0305004100056887
John Martino and Stewart Priddy, Classification of $BG$ for groups with dihedral or quarternion Sylow $2$-subgroups, J. Pure Appl. Algebra 73 (1991), no. 1, 13–21. MR 1121628, DOI 10.1016/0022-4049(91)90103-9
John Martino and Stewart Priddy, The complete stable splitting for the classifying space of a finite group, Topology 31 (1992), no. 1, 143–156. MR 1153242, DOI 10.1016/0040-9383(92)90067-R
John Martino and Stewart Priddy, A classification of the stable type of $BG$, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 165–170. MR 1145578, DOI 10.1090/S0273-0979-1992-00300-2
T. Venkatarayudu, The $7$-$15$ problem, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 531. MR 0000001, DOI 10.1090/gsm/058
Haynes Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. (2) 120 (1984), no. 1, 39–87. MR 750716, DOI 10.2307/2007071
Stephen A. Mitchell, Splitting $B(\textbf {Z}/p)^n$ and $BT^n$ via modular representation theory, Math. Z. 189 (1985), no. 1, 1–9. MR 776532, DOI 10.1007/BF01246939
Jesse Douglas, The most general form of the problem of Plateau, Amer. J. Math. 61 (1939), 590–608. MR 102, DOI 10.2307/2371315
Stewart B. Priddy, On characterizing summands in the classifying space of a group. I, Amer. J. Math. 112 (1990), no. 5, 737–748. MR 1073007, DOI 10.2307/2374805
Stewart B. Priddy, On characterizing summands in the classifying space of a group. II, Homotopy theory and related topics (Kinosaki, 1988) Lecture Notes in Math., vol. 1418, Springer, Berlin, 1990, pp. 175–183. MR 1048185, DOI 10.1007/BFb0083702
Douglas C. Ravenel, The Segal conjecture for cyclic groups, Bull. London Math. Soc. 13 (1981), no. 1, 42–44. MR 599639, DOI 10.1112/blms/13.1.42
C. Witten. Self-maps of classifying spaces of finite groups and classification of low-dimensional Poincaré duality spaces. Ph. D. Thesis, Stanford University, 1978.