Fusion Systems in Algebra and Topology

by Aschbacher, Michael; Kessar, Radha; Oliver, Bob

• ISBN13: 9781107601000
• ISBN10: 1107601002
• Edition: 1st
• Format: Paperback
• Copyright: 2011-10-31
• Publisher: Cambridge Univ Pr

Summary

A fusion system over a p-group S is a category whose objects form the set of all subgroups of S, whose morphisms are certain injective group homomorphisms, and which satisfies axioms first formulated by Puig that are modelled on conjugacy relations in finite groups. The definition was originally motivated by representation theory, but fusion systems also have applications to local group theory and to homotopy theory. The connection with homotopy theory arises through classifying spaces which can be associated to fusion systems and which have many of the nice properties of p-completed classifying spaces of finite groups. Beginning with a detailed exposition of the foundational material, the authors then proceed to discuss the role of fusion systems in local finite group theory, homotopy theory and modular representation theory. The book serves as a basic reference and as an introduction to the field, particularly for students and other young mathematicians.

Table of Contents

Introduction	p. 1
Introduction to fusion systems	p. 5
The fusion category of a finite group	p. 5
Abstract fusion systems	p. 7
Alperin's fusion theorem	p. 11
Normal and central subgroups of a fusion system	p. 17
Normalizer fusion systems	p. 21
Normal fusion subsystems and products	p. 25
Fusion subsystems of p-power index or of index prime to p	p. 32
The transfer homomorphism for saturated fusion systems	p. 38
Other definitions of saturation	p. 45
The local theory of fusion systems	p. 49
Notation and terminology on groups	p. 51
Fusion systems	p. 51
Saturated fusion systems	p. 53
Models for constrained saturated fusion systems	p. 54
Factor systems and surjective morphisms	p. 56
Invariant subsystems of fusion systems	p. 60
Normal subsystems of fusion systems	p. 62
Invariant maps and normal maps	p. 65
Theorems on normal subsystems	p. 68
Composition series	p. 71
Constrained systems	p. 78
Solvable systems	p. 80
Fusion systems in simple groups	p. 84
Classifying simple groups and fusion systems	p. 87
Systems of characteristic 2-type	p. 93
Fusion and homotopy theory	p. 103
Classifying spaces, p-completion, and the Martino-Priddy conjecture	p. 106
Homotopy and fundamental groups	p. 106
CW complexes and cellular homology	p. 109
Classifying spaces of discrete groups	p. 110
The p-completion functor of Bousfield and Kan	p. 113
Equivalences between fusion systems of finite groups	p. 116
The Martino-Priddy conjecture	p. 117
An application: fusion in finite groups of Lie type	p. 118
The geometric realization of a category	p. 119
Simplicial sets and their realizations	p. 120
The nerve of a category as a simplicial set	p. 122
Classifying spaces as geometric realizations of categories	p. 124
Fundamental groups and coverings of geometric realizations	p. 125
Spaces of maps	p. 129
Linking systems and classifying spaces of finite groups	p. 133
The linking category of a finite group	p. 133
Fusion and linking categories of spaces	p. 135
Linking systems and equivalences of p-completed classifying spaces	p. 138
Abstract fusion and linking systems	p. 139
Linking systems, centric linking systems and p-local finite groups	p. 140
Quasicentric subgroups and quasicentric linking systems	p. 143
Automorphisms of fusion and linking systems	p. 152
Normal fusion and linking subsystems	p. 155
Fundamental groups and covering spaces	p. 158
Homotopy properties of classifying spaces	p. 161
Classifying spectra of fusion systems	p. 165
An infinite version: p-local compact groups	p. 167
The orbit category and its applications	p. 168
Higher limits of functors and the bar resolution	p. 170
Constrained fusion systems	p. 175
Existence, uniqueness, and automorphisms of linking systems	p. 182
Some computational techniques for higher limits over orbit categories	p. 189
Homotopy colimits and homotopy decompositions	p. 197
The subgroup decomposition of L	p. 200
An outline of the proofs of Theorems 4.21 and 4.22	p. 204
The centralizer and normalizer decompositions of L	p. 207
Examples of exotic fusion systems	p. 209
Reduced fusion systems and tame fusion systems	p. 210
The Ruiz-Viruel examples	p. 212
Saturated fusion systems over 2-groups	p. 214
Mixing related fusion systems	p. 215
Other examples	p. 215
Open problems	p. 216
Fusion and Representation theory	p. 220
Algebras and G-algebras	p. 222
Ideals and Idempotents	p. 222
G-algebras	p. 226
Relative trace maps and Brauer homomorphisms	p. 227
p-permutation algebras, Brauer pairs and fusion systems	p. 232
p-permutation algebras and the Brauer homomorphisms	p. 232
(A, G)-Brauer pairs and inclusion.	p. 235
(A, b, G)-Brauer pairs and inclusion.	p. 241
(A, b, G)-Brauer pairs and fusion systems	p. 243
p-permutation algebras and saturated fusion systems	p. 244
Saturated triples	p. 244
Normaliser systems and saturated triples	p. 249
Saturated triples and normal subgroups	p. 251
Block fusion systems	p. 253
Fusion systems of blocks of local subgroups	p. 255
Background on finite group representations	p. 258
Ordinary and modular representations	p. 259
p-modular systems	p. 261
Cartan and decomposition maps	p. 262
Ordinary and Brauer characters	p. 266
Fusion and structure	p. 270
The three main theorems of Brauer	p. 270
Relative projectivity and representation type	p. 274
Finiteness conjectures	p. 276
Source algebras and Puig's conjecture	p. 278
Külshammer-Puig classes	p. 281
Nilpotent blocks and extensions	p. 286
Counting Conjectures	p. 288
Block fusion systems and normal subgroups.	p. 293
Open Problems	p. 298
Background facts about groups	p. 300
References	p. 306
List of notation	p. 314
Index	p. 317
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