Classifying spectra of saturated fusion systems

Ragnarsson, Kári¹

¹Department of Mathematical Sciences, University of Aberdeen, Aberdeen AB24 3UE, UK

Algebraic and Geometric Topology, Tome 6 (2006) no. 1, pp. 195-252

Résumé

The assignment of classifying spectra to saturated fusion systems was suggested by Linckelmann and Webb and has been carried out by Broto, Levi and Oliver. A more rigid (but equivalent) construction of the classifying spectra is given in this paper. It is shown that the assignment is functorial for fusion-preserving homomorphisms in a way which extends the assignment of stable p–completed classifying spaces to finite groups, and admits a transfer theory analogous to that for finite groups. Furthermore the group of homotopy classes of maps between classifying spectra is described, and in particular it is shown that a fusion system can be reconstructed from its classifying spectrum regarded as an object under the stable classifying space of the underlying p–group.

Détail

DOI : 10.2140/agt.2006.6.195

Keywords: fusion systems, p-local finite groups, stable homotopy, transfer

Bibliographie

[1] J F Adams, Stable homotopy and generalised homology, University of Chicago Press (1974)

[2] J F Adams, Infinite loop spaces, Annals of Mathematics Studies 90, Princeton University Press (1978)

[3] J C Becker, D H Gottlieb, The transfer map and fiber bundles, Topology 14 (1975) 1

[4] D J Benson, M Feshbach, Stable splittings of classifying spaces of finite groups, Topology 31 (1992) 157

[5] A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Springer (1972)

[6] C Broto, R Levi, B Oliver, Homotopy equivalences of $p$–completed classifying spaces of finite groups, Invent. Math. 151 (2003) 611

[7] C Broto, R Levi, B Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003) 779

[8] G Carlsson, Equivariant stable homotopy and Segal's Burnside ring conjecture, Ann. of Math. $(2)$ 120 (1984) 189

[9] N Castellana, L Morales, Vector bundles over classifying spaces revisited, in preparation

[10] W G Dwyer, Transfer maps for fibrations, Math. Proc. Cambridge Philos. Soc. 120 (1996) 221

[11] R Kessar, R Stancu, A reduction theorem for fusion systems of blocks, preprint

[12] L G Lewis, J P May, J E Mcclure, Classifying $G$–spaces and the Segal conjecture, from: "Current trends in algebraic topology, Part 2 (London, Ont., 1981)", CMS Conf. Proc. 2, Amer. Math. Soc. (1982) 165

[13] M Linckelmann, P Webb, unpublished work,

[14] J Martino, S Priddy, Stable homotopy classification of $BG_{p}^{\wedge}$, Topology 34 (1995) 633

[15] J Martino, S Priddy, Unstable homotopy classification of $BG_{p}^{\wedge}$, Math. Proc. Cambridge Philos. Soc. 119 (1996) 119

[16] J P May, J E Mcclure, A reduction of the Segal conjecture, from: "Current trends in algebraic topology, Part 2 (London, Ont., 1981)", CMS Conf. Proc. 2, Amer. Math. Soc. (1982) 209

[17] G Nishida, Stable homotopy type of classifying spaces of finite groups, from: "Algebraic and topological theories (Kinosaki, 1984)", Kinokuniya (1986) 391

[18] B Oliver, Equivalences of classifying spaces completed at the prime two, Mem. Amer. Math. Soc. (to appear)

[19] B Oliver, Equivalences of classifying spaces completed at odd primes, Math. Proc. Cambridge Philos. Soc. 137 (2004) 321

[20] L Puig, Frobenius systems and their localizing categories, preprint

[21] L Puig, unpublished notes,

[22] K Ragnarsson, Alternative stable homotopy classification of $BG_{p}^{\wedge}$, Topology (to appear)

[23] K Ragnarsson, Retractive transfers and $p$–local finite groups, preprint

[24] K Ragnarsson, A Segal conjecture for $p$–completed classifying spaces, preprint

[25] T Tom Dieck, Transformation groups and representation theory, Lecture Notes in Mathematics 766, Springer (1979)