Algebra & Number Theory

The abelian monoid of fusion-stable finite sets is free

Sune Reeh

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/ant.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We show that the abelian monoid of isomorphism classes of G-stable finite S-sets is free for a finite group G with Sylow p-subgroup S; here a finite S-set is called G-stable if it has isomorphic restrictions to G-conjugate subgroups of S. These G-stable S-sets are of interest, e.g., in homotopy theory. We prove freeness by constructing an explicit (but somewhat nonobvious) basis, whose elements are in one-to-one correspondence with the G-conjugacy classes of subgroups in S. As a central tool of independent interest, we give a detailed description of the embedding of the Burnside ring for a saturated fusion system into its associated ghost ring.

Article information

Source
Algebra Number Theory, Volume 9, Number 10 (2015), 2303-2324.

Dates
Received: 3 December 2014
Revised: 31 August 2015
Accepted: 8 October 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842447

Digital Object Identifier
doi:10.2140/ant.2015.9.2303

Mathematical Reviews number (MathSciNet)
MR3437763

Zentralblatt MATH identifier
1369.20024

Subjects
Primary: 20D20: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure
Secondary: 20J15: Category of groups 19A22: Frobenius induction, Burnside and representation rings

Keywords
Fusion systems Burnside rings finite groups

Citation

Reeh, Sune. The abelian monoid of fusion-stable finite sets is free. Algebra Number Theory 9 (2015), no. 10, 2303--2324. doi:10.2140/ant.2015.9.2303. https://projecteuclid.org/euclid.ant/1510842447


Export citation

References

  • M. Aschbacher, R. Kessar, and B. Oliver, Fusion systems in algebra and topology, London Mathematical Society Lecture Note Series 391, Cambridge University Press, 2011.
  • A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, vol. 304, Lecture Notes in Mathematics, Springer, Berlin, 1972.
  • C. Broto, R. Levi, and B. Oliver, “The homotopy theory of fusion systems”, J. Amer. Math. Soc. 16:4 (2003), 779–856.
  • G. Carlsson, “Equivariant stable homotopy and Sullivan's conjecture”, Invent. Math. 103:3 (1991), 497–525.
  • A. Díaz and A. Libman, “The Burnside ring of fusion systems”, Adv. Math. 222:6 (2009), 1943–1963.
  • A. Díaz and A. Libman, “Segal's conjecture and the Burnside rings of fusion systems”, J. Lond. Math. Soc. $(2)$ 80:3 (2009), 665–679.
  • T. tom Dieck, Transformation groups and representation theory, Lecture Notes in Mathematics 766, Springer, Berlin, 1979.
  • A. W. M. Dress, “Congruence relations characterizing the representation ring of the symmetric group”, J. Algebra 101:2 (1986), 350–364.
  • W. Dwyer and A. Zabrodsky, “Maps between classifying spaces”, pp. 106–119 in Algebraic topology, Barcelona, 1986, edited by J. Aguadé and R. Kane, Lecture Notes in Math. 1298, Springer, Berlin, 1987.
  • M. J. K. Gelvin, Fusion action systems, Ph.D. thesis, Massachusetts Institute of Technology, 2010, hook http://search.proquest.com/docview/847033297 \posturlhook.
  • J. Grodal, “Group actions on sets, at a prime $p$”, pp. 2647–2648 in Homotopy theory, Oberwolfach Rep. 8 :3, European Mathematican Society, Zürich, 2011.
  • J. Grodal, “The Burnside ring of the $p$-completed classifying space of a finite group”, in preparation.
  • J. Lannes, “Applications dont la source est un classifiant”, pp. 566–573 in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), edited by S. D. Chatterji, Birkhäuser, Basel, 1995.
  • H. Miller, “The Sullivan conjecture and homotopical representation theory”, pp. 580–589 in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), edited by A. M. Gleason, Amer. Math. Soc., Providence, RI, 1987.
  • G. Mislin, “On group homomorphisms inducing mod-$p$ cohomology isomorphisms”, Comment. Math. Helv. 65:3 (1990), 454–461.
  • K. Ragnarsson and R. Stancu, “Saturated fusion systems as idempotents in the double Burnside ring”, Geom. Topol. 17:2 (2013), 839–904.
  • S. P. Reeh, “Transfer and characteristic idempotents for saturated fusion systems”, Adv. in Math. 289 (2016), 161–211.
  • K. Roberts and S. Shpectorov, “On the definition of saturated fusion systems”, J. Group Theory 12:5 (2009), 679–687.
  • T. Yoshida, “On the unit groups of Burnside rings”, J. Math. Soc. Japan 42:1 (1990), 31–64.