Existence and uniqueness of classifying spaces for fusion systems over discrete \(p\)-toral groups. (English) Zbl 1346.55015
The first open problem of Oliver’s list of open problems in the homotopy theoretic side of fusion systems was whether for every saturated fusion system over a finite \(p\)-group there exists an associated centric linking system and if so whether it is unique. This was resolved positively by A. Chermak [Acta Math. 211, No. 1, 47–139 (2013; Zbl 1295.20021)] using partial groups and localities. The proof was translated by Oliver into obstruction theory. The authors extend Oliver’s proof to the \(p\)-local compact case. This extends the result of existence and uniqueness of an associated centric linking to the case of saturated fusion systems over discrete \(p\)-toral groups.
Reviewer: Nora Seeliger (Canberra)
MSC:
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
| 20J05 | Homological methods in group theory |
| 20N99 | Other generalizations of groups |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
Keywords:
classifying space; fusion systems; localities; higher limits; obstruction theory; best offenders; Thompson groupCitations:
Zbl 1295.20021References:
| [1] | M.Aschbacher, R.Kessar, B.Oliver, Fusion systems in algebra and topology (Cambridge University Press, Cambridge, 2011). · Zbl 1255.20001 |
| [2] | C.Broto, N.Castellana, J.Grodal, R.Levi, B.Oliver, ‘Subgroup families controlling p-local finite groups’, Proc. London Math. Soc., (3)91 (2005) 325-354. · Zbl 1090.20026 |
| [3] | C.Broto, R.Levi, B.Oliver, ‘The homotopy theory of fusion systems’, J. Amer. Math. Soc., 16 (2003) 779-856. · Zbl 1033.55010 |
| [4] | C.Broto, R.Levi, B.Oliver, ‘Discrete models for the \(p\)‐local homotopy theory of compact Lie groups and \(p\)‐compact groups’, Geom. Topol., 11 (2007) 315-427. · Zbl 1135.55008 |
| [5] | C.Broto, R.Levi, B.Oliver, ‘An algebraic model for finite loop spaces’, Algeb. Geom. Topol., 14 (2014) 2915-2982. · Zbl 1306.55008 |
| [6] | H.Cartan, S.Eilenberg, Homological algebra (Princeton University Press, Princeton, NJ, 1956). · Zbl 0075.24305 |
| [7] | A.Chermak, ‘Fusion systems and localities’, Acta Math., 211 (2013) 47-139. · Zbl 1295.20021 |
| [8] | L.Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics 36 (Academic Press, New York-London, 1970). · Zbl 0209.05503 |
| [9] | A.Gonzalez, ‘The structure of \(p\)‐local compact groups’, PhD Thesis, Universitat Autonoma de Barcelona, 2010. |
| [10] | S.Jackowski, J.McClure, B.Oliver, ‘Homotopy classification of self-maps of \(B G\) via \(G\)-actions. II’, Ann. of Math., (2)135 (1992) 227-270. · Zbl 0771.55003 |
| [11] | U.Meierfrankenfeld, B.Stellmacher, ‘The general FF-module theorem’, J. Algebra, 351 (2012) 1-63. · Zbl 1262.20003 |
| [12] | B.Oliver, ‘Existence and uniqueness of linking systems: Chermak’s proof via obstruction theory’, Acta Math., 211 (2013) 141-175. · Zbl 1292.55007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
