Department of Mathematical Sciences > Research > Conferences > 2010 and before > LocalFS
Local methods in fusion systemsThe University of CopenhagenDecember 1519, 2008 
Organizer: Antonio Díaz Ramos
External Participants:
 Adam Glesser (Suffolk University)
 Nadia Mazza (University of Lancaster)
 Sejong Park (Oberwolfach Institute)
 Radu Stancu (Universite de Picardie)
This is a workshop on fusion systems. There will be very few talks (1 per day) and plenty of time to interact. The interest is to make further progress in generalizing results from group theory to the fusion system setting. Nevertheless, the talks are not restricted to this topic, and they cover several topics related to fusion systems.
Programme:
Mon 15  13:1514:15  Aud. 2 
Antonio Díaz 
Introduction to fusion systems 
Tue 16  13:1514:15  Aud. 2 
Sejong Park 
Control of fusion and transfer 
Wed 17  10:1511:15  Aud. 9 
Nadia Mazza 
Oliver's conjecture 
Thu 18  13:1514:15  Aud. 9 
Radu Stancu 
Characteristic bisets and fusion 
Fri 19  10:1511:15  Aud. 8 
Adam Glesser 
Sparse fusion systems 
Overview of the topics:

Introduction to fusion systems: this will be an introduction to fusion systems theory with emphasis on the homotopy theoretical point of view. Main open problems of the theory will be described. Analogies with group theory will be discussed (recommended reading: [BLO04]).

Sparse fusion systems: In studying fusion systems, one is inexorably led towards proving statements by considering a minimal counterexample and showing that the accompanying fusion system is constrained, hence modeled by a finite group. Commonly, the fusion system involved has very few subfusion systems. In an extreme case, we call the fusion system sparse. In this talk, we will give some basic properties of sparse fusion systems, allowing us to streamline the proof of a result of Kessar and Linckelmann as well as to give a proof of a new result based on an unpublished lemma of Navarro.

Oliver's conjecture: In the proof of the MartinoPriddy conjecture in odd characteristic Bob Oliver defines a certain characteristic subgroup in any finite pgroup (Definition 3.1 in [O04]). He then conjectures that this subgroup always contains the Thompson subgroup that is built up using the elementary abelian subgroups of maximal order (Conjecture 3.9 in [O04]). In fact, if it were true, then it would provide another proof of the MartinoPriddy conjecture in odd characteristic. But a proof of Oliver's conjecture is still awaiting, and only few cases have been shown. In this talk, we will present a survey of the conjecture and an update of the results known so far (recommended reading: [O04,GHL]).

Control of fusion and transfer: We review the notion of control of fusion and transfer for finite groups and see how it can be generalized for (saturated) fusion systems. Then we discuss recent generalizations of classical control of fusion/transfer theorems in local group theory to fusion systems  notably GlaubermanThompson pnilpotency theorem and Glauberman's ZJ theorem [KL], Thompson's pnilpotency theorem [DGMP1], Stellmacher's characteristic 2subgroup theorem [OS], and Glauberman's Kgroup theorem [KL] [DGMP2]. Finally we discuss a new generalization of Yoshida's theorem on control of transfer to fusion systems.

Characteristic bisets and fusion: To a saturated fusion system on a finite $p$group $S$ Broto, Levi and Oliver associate a characteristic $S$$S$ biset having stability properties with respect to the fusion system. I'll explain that the existence of such characteristic biset implies the saturation of the fusion system. A characteristic biset associated to a saturated fusion system also satisfies some Frobenius reciprocity type properties (that I'll introduce and discuss). Again we have a converse statement saying that if a characteristic biset of a fusion system satisfies Frobenius reciprocity then the fusion system is saturated. This is joint work with Kari Ragnarsson.

Generalizing results from group theory to fusion systems: Alperin's fusion theorem for finite groups is the most central result which has successfully been generalized to the fusion system setting. Other such results are Frobenius' theorem on normal pcomplements, Glauberman and Thompson's pnilpotency criterion and Glauberman's ZJtheorem.
We will use the visitors office 4.01 for informal meetings. The discussion topics in these meeting will be all of the above and possibly:

Yoshida's theorem

Category of elementary abelian psubgroups

Burnside ring of fusion systems
Bibliography:

[BLO04] C. Broto, R. Levi, R. Oliver, The theory of plocal groups: a survey, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic Ktheory, 5184, Contemp. Math., 346, Amer. Math. Soc., Providence, RI, 2004.

[GHL] D.J. Green, L. Hethelyi, and M. Lilienthal, On Oliver's pgroup conjecture, preprint.

[DGMP1] A. Diaz, A. Glesser, N. Mazza, S. Park, Glauberman and Thompson's theorems for fusion systems, Proc. Amer. Math. Soc., 137 (2009), 495503.

[DGMP2] A. Diaz, A. Glesser, N. Mazza, S. Park, Control of transfer
and weak closure in fusion systems, preprint (2008), preprint. 
[KL] R. Kessar, M. Linckelmann, ZJtheorem for fusion systems, Trans. Amer. Math. Soc., 360 (2008), 30933206.

[O04] R. Oliver, Equivalences of classifying spaces completed at odd primes, Math. Proc. Cambridge Philos. Soc., vol. 137 (2004), number 2, 321347.

[OS] S. Onofrei, R. Stancu, A characteristic subgroup for fusion systems, preprint (2008), preprint.