Geometry & Topology

Saturated fusion systems as idempotents in the double Burnside ring

Kári Ragnarsson and Radu Stancu

Full-text: Open access


We give a new characterization of saturated fusion systems on a p–group S in terms of idempotents in the p–local double Burnside ring of S that satisfy a Frobenius reciprocity relation. Interpreting our results in stable homotopy, we characterize the stable summands of the classifying space of a finite p–group that have the homotopy type of the classifying spectrum of a saturated fusion system, and prove an invariant theorem for double Burnside modules analogous to the Adams–Wilkerson criterion for rings of invariants in the cohomology of an elementary abelian p–group. This work is partly motivated by a conjecture of Haynes Miller that proposes p–tract groups as a purely homotopy-theoretical model for p–local finite groups. We show that a p–tract group gives rise to a p–local finite group when two technical assumptions are made, thus reducing the conjecture to proving those two assumptions.

Article information

Geom. Topol., Volume 17, Number 2 (2013), 839-904.

Received: 23 December 2010
Revised: 7 May 2012
Accepted: 12 December 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20D20: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure 55R35: Classifying spaces of groups and $H$-spaces
Secondary: 55P42: Stable homotopy theory, spectra 19A22: Frobenius induction, Burnside and representation rings

fusion system Burnside ring finite groups classifying spaces stable splitting


Ragnarsson, Kári; Stancu, Radu. Saturated fusion systems as idempotents in the double Burnside ring. Geom. Topol. 17 (2013), no. 2, 839--904. doi:10.2140/gt.2013.17.839.

Export citation


  • J F Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics, University of Chicago Press (1974)
  • J F Adams, Infinite loop spaces, Annals of Mathematics Studies 90, Princeton Univ. Press (1978)
  • J F Adams, C W Wilkerson, Finite $H$-spaces and algebras over the Steenrod algebra, Ann. of Math. 111 (1980) 95–143
  • J Alperin, M Broué, Local methods in block theory, Ann. of Math. 110 (1979) 143–157
  • M Aschbacher, Normal subsystems of fusion systems, Proc. Lond. Math. Soc. 97 (2008) 239–271
  • M Aschbacher, Generation of fusion systems of characteristic 2-type, Invent. Math. 180 (2010) 225–299
  • M Aschbacher, The generalized Fitting subsystem of a fusion system, Mem. Amer. Math. Soc. 209 (2011) vi+110
  • M Aschbacher, $S_3$–free $2$–fusion systems, Proc. Edinb. Mth. Soc. 56 (2013) 27–48
  • M Aschbacher, A Chermak, A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver, Ann. of Math. 171 (2010) 881–978
  • M F Atiyah, I G Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, MA (1969)
  • D Benson, Stably splitting $BG$, Bull. Amer. Math. Soc. 33 (1996) 189–198
  • D J Benson, M Feshbach, Stable splittings of classifying spaces of finite groups, Topology 31 (1992) 157–176
  • R Boltje, S Danz, A ghost ring for the left-free double Burnside ring and an application to fusion systems, Adv. Math. 229 (2012) 1688–1733
  • S Bouc, Burnside rings, from: “Handbook of algebra, Vol. 2”, (M Hazewinkel, editor), North-Holland, Amsterdam (2000) 739–804
  • C Broto, N Castellana, J Grodal, R Levi, B Oliver, Extensions of $p$-local finite groups, Trans. Amer. Math. Soc. 359 (2007) 3791–3858
  • C Broto, R Levi, B Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003) 779–856
  • C Broto, R Levi, B Oliver, A geometric construction of saturated fusion systems, from: “An alpine anthology of homotopy theory”, (D Arlettaz, K Hess, editors), Contemp. Math. 399, Amer. Math. Soc., Providence, RI (2006) 11–40
  • C Broto, J M Møller, Chevalley $p$-local finite groups, Algebr. Geom. Topol. 7 (2007) 1809–1919
  • G Carlsson, Equivariant stable homotopy and Segal's Burnside ring conjecture, Ann. of Math. 120 (1984) 189–224
  • A Chermak, Fusion systems and localities, preprint (2011) Available at \setbox0\makeatletter\@url {\unhbox0
  • A Díaz, A Libman, Segal's conjecture and the Burnside rings of fusion systems, J. Lond. Math. Soc. 80 (2009) 665–679
  • M Gelvin, K Ragnarsson, A homotopy characterization of $p$–completed classifying spaces of finite groups, in preparation (2011)
  • P Goerss, L Smith, S Zarati, Sur les $A$-algèbres instables, from: “Algebraic topology, Barcelona, 1986”, (J Aguadé, R Kane, editors), Lecture Notes in Math. 1298, Springer, Berlin (1987) 148–161
  • R Kessar, R Stancu, A reduction theorem for fusion systems of blocks, J. Algebra 319 (2008) 806–823
  • L G Lewis, J P May, J E McClure, Classifying $G$-spaces and the Segal conjecture, from: “Current trends in algebraic topology, Part 2”, (R M Kane, S O Kochman, P S Selick, V P Snaith, editors), CMS Conf. Proc. 2, Amer. Math. Soc., Providence, R.I. (1982) 165–179
  • M Linckelmann, Alperin's weight conjecture in terms of equivariant Bredon cohomology, Math. Z. 250 (2005) 495–513
  • J Martino, S Priddy, The complete stable splitting for the classifying space of a finite group, Topology 31 (1992) 143–156
  • J P May, J E McClure, A reduction of the Segal conjecture, from: “Current trends in algebraic topology, Part 2”, (R M Kane, S O Kochman, P S Selick, V P Snaith, editors), CMS Conf. Proc. 2, Amer. Math. Soc., Providence, R.I. (1982) 209–222
  • H Miller, Massey–Peterson towers and maps from classifying spaces, from: “Algebraic topology”, (I Madsen, B Oliver, editors), Lecture Notes in Math. 1051, Springer, Berlin (1984) 401–417
  • G Nishida, Stable homotopy type of classifying spaces of finite groups, from: “Algebraic and topological theories”, (M Nagata, S Araki, A Hattori, N Iwahori, et al, editors), Kinokuniya, Tokyo (1986) 391–404
  • B Oliver, Existence and uniqueness of linking systems: Chermak's proof via obstruction theory
  • L Puig, Frobenius categories, J. Algebra 303 (2006) 309–357
  • L Puig, Frobenius categories versus Brauer blocks: the Grothendieck group of the Frobenius category of a Brauer block, Progress in Mathematics 274, Birkhäuser, Basel (2009)
  • K Ragnarsson, Classifying spectra of saturated fusion systems, Algebr. Geom. Topol. 6 (2006) 195–252
  • K Ragnarsson, Retractive transfers and $p$-local finite groups, Proc. Edinb. Math. Soc. 51 (2008) 465–487
  • S Reeh, Burnside rings of fusion systems, MSc thesis, University of Copenhagen (2010)
  • K Roberts, S Shpectorov, On the definition of saturated fusion systems, J. Group Theory 12 (2009) 679–687