Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Stably splitting BG

Author: Dave Benson
Journal: Bull. Amer. Math. Soc. 33 (1996), 189-198
MSC (1991): Primary 55P
MathSciNet review: 1362628
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. J. F. Adams. Graeme Segal's Burnside ring conjecture. Contemp. Math. 12 (1982), 9--18. MR 84b:55024
  • 2. J. F. Adams, J. H. C. Gunawardena and H. Miller. The Segal conjecture for elementary abelian $p$-groups, I. Topology 24 (1985), 435--460. MR 87m:55026
  • 3. M. Atiyah. Characters and cohomology of finite groups. Inst. Hautes Études Sci. Publ. Math. 9 (1961), 23--64. MR 26:6228
  • 4. D. J. Benson and M. Feshbach. Stable splittings of classifying spaces of finite groups. Topology 31 (1992), 157--176. MR 93d:55013
  • 5. D. Carlisle and N. Kuhn. Subalgebras of the Steenrod algebra and the action of matrices on truncated polynomial algebras. J. Algebra 121 (1989), 370--387. MR 90c:55018
  • 6. G. Carlsson. G. B. Segal's Burnside ring conjecture for $(\mathbb Z/2\mathbb Z)^k$. Topology 22 (1983), 83--103. MR 84a:55007
  • 7. G. Carlsson. Equivariant stable homotopy and Segal's Burnside ring conjecture. Annals of Math. 120 (1984), 189--224. MR 86f:57036
  • 8. A. H. Clifford and G. B. Preston. Algebraic Theory of Semigroups, I. Mathematical Surveys of the A.M.S. 7, 1961. MR 24:A2627
  • 9. C. W. Curtis and I. Reiner. Methods of Representation Theory, I. Wiley-Interscience, 1981. MR 82i:20001
  • 10. E. Devinatz, M. J. Hopkins and J. H. Smith. Nilpotence and stable homotopy theory. Annals of Math. 128 (1988), 207--242. MR 89m:55009
  • 11. J. Dietz. Stable splittings of classifying spaces of metacyclic $p$-groups, $p$ odd., J. Pure & Applied Algebra 90 (1993), 115--136. MR 95f:55014
  • 12. J. Dietz and S. B. Priddy. The stable homotopy type of rank two $p$-groups. Contemp. Math. vol. 188, Amer. Math. Soc., Providence, RI, 1995. MR 1:349 132
  • 13. J. H. C. Gunawardena. Segal's conjecture for cyclic groups of (odd) prime order. J. T. Knight Prize Essay, Cambridge, 1980.
  • 14. J. C. Harris. Thesis. University of Chicago, 1985.
  • 15. J. C. Harris and N. J. Kuhn. Stable decompositions of classifying spaces of finite abelian $p$- groups. Math. Proc. Camb. Phil. Soc. 103 (1988), 427--449. MR 89d:55021
  • 16. J. Howie. An Introduction to Semigroup Theory. Academic Press, London 1976. MR 57:6235
  • 17. J. Lannes. Sur les espaces fonctionnels dont la source est le classifiant d'un $p$-groupe abélien élémentaire. Publ. Math. IHES 75 (1992), 135--244. MR 93j:55019
  • 18. G. Lewis, J. P. May and J. E. McClure. Classifying $G$-spaces and the Segal conjecture. Current Trends in Algebraic Topology. CMS Conference Proceedings 2 (1981), 165--179. MR 84d:55007a
  • 19. W. H. Lin. On conjectures of Mahowald, Segal and Sullivan. Math. Proc. Camb. Phil. Soc. 87 (1980), 449--458. MR 81e:55020
  • 20. J. Martino and S. B. Priddy. Classification of $BG$ for groups with dihedral or quaternion Sylow $2$- subgroups. J. Pure & Applied Algebra 73 (1991), 13--21. MR 92f:55022
  • 21. J. Martino and S. B. Priddy. The complete stable splitting for the classifying space of a finite group. Topology 31 (1992), 143--156. MR 93d:55012
  • 22. J. Martino and S. B. Priddy. A classification of the stable type of $BG$. Bull. of the A.M.S. 27 (1992), 165--170. MR 93b:55019
  • 23. J. Martino and S. B. Priddy. Stable homotopy classification of $BG^{\hat{\ }}_p$. Topology 34 (1995), 633--649. MR 1:341 812
  • 24. H. Miller. The Sullivan conjecture on maps from classifying spaces. Ann. of Math. 120 (1984), 39--87. MR 85i:55012
  • 25. S. Mitchell. Splitting $B(\mathbb Z/p)^n$ and $BT^n$ via modular representation theory. Math. Zeit. 189 (1985), 1--9. MR 86i:55020
  • 26. G. Nishida. Stable homotopy type of classfying spaces of finite groups. Algebraic and Topological Theories (1985), 391--404. MR 102:269
  • 27. S. B. Priddy. On characterizing summands in the classfying space of a finite group, I. Amer. J. Math. 112 (1990), 737--748. MR 91i:55020
  • 28. S. B. Priddy. On characterizing summands in the classfying space of a finite group, II. Homotopy Theory and Related Topics. Springer Lecture Notes in Mathematics 1418 (1990). MR 91j:55010
  • 29. D. Ravenel. The Segal conjecture for cyclic groups. Bull. London Math. Soc. 13 (1981), 42--44. MR 82e:55017
  • 30. C. Witten. Self-maps of classifying spaces of finite groups and classification of low-dimensional Poincaré duality spaces. Ph. D. Thesis, Stanford University, 1978.

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1991): 55P

Retrieve articles in all journals with MSC (1991): 55P

Additional Information

Dave Benson
Affiliation: Department of Mathematics, University of Georgia, Athens GA 30602, USA

Keywords: Stable homotopy, stable splitting, classifying space, double Burnside ring
Additional Notes: Partly supported by a grant from the NSF
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society