Fusion Systems in Algebra and Topology

Cover......Page 1
LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES......Page 2
Title......Page 4
Copyright......Page 5
Contents......Page 6
Introduction......Page 10
1. The fusion category of a finite group......Page 14
2. Abstract fusion systems......Page 16
3. Alperin’s fusion theorem......Page 20
4. Normal and central subgroups of a fusion system......Page 26
5. Normalizer fusion systems......Page 30
6. Normal fusion subsystems and products......Page 34
7. Fusion subsystems of p-power index or of index prime to p......Page 41
8. The transfer homomorphism for saturated fusion systems......Page 47
9. Other definitions of saturation......Page 54
Part II. The local theory of fusion systems......Page 58
2. Fusion systems......Page 60
Exercises for Section 2......Page 61
3. Saturated fusion systems......Page 62
4. Models for constrained saturated fusion systems......Page 63
Exercises for Section 4......Page 64
5. Factor systems and surjective morphisms......Page 65
6. Invariant subsystems of fusion systems......Page 69
7. Normal subsystems of fusion systems......Page 71
8. Invariant maps and normal maps......Page 74
Exercises for Section 8......Page 76
9. Theorems on normal subsystems......Page 77
10. Composition series......Page 80
Exercises for Section 10......Page 86
11. Constrained systems......Page 87
12. Solvable systems......Page 89
13. Fusion systems in simple groups......Page 93
14. Classifying simple groups and fusion systems......Page 96
15. Systems of characteristic 2-type......Page 102
Exercises for Section 15......Page 111
Part III. Fusion and homotopy theory......Page 112
1.1. Homotopy and fundamental groups.......Page 115
1.2. CW complexes and cellular homology.......Page 118
1.3. Classifying spaces of discrete groups.......Page 119
1.4. The p-completion functor of Bousfield and Kan.......Page 122
1.5. Equivalences between fusion systems of finite groups.......Page 125
1.6. The Martino-Priddy conjecture.......Page 126
1.7. An application: fusion in finite groups of Lie type.......Page 127
2. The geometric realization of a category......Page 128
2.1. Simplicial sets and their realizations.......Page 129
2.2. The nerve of a category as a simplicial set.......Page 131
2.3. Classifying spaces as geometric realizations of categories.......Page 133
2.4. Fundamental groups and coverings of geometric realizations.......Page 134

2.5. Spaces of maps.......Page 138
3.1. The linking category of a finite group.......Page 142
3.2. Fusion and linking categories of spaces.......Page 144
3.3. Linking systems and equivalences of p-completed classifying spaces.......Page 147
4. Abstract fusion and linking systems......Page 148
4.1. Linking systems, centric linking systems and p-local finite groups.......Page 149
4.2. Quasicentric subgroups and quasicentric linking systems.......Page 152
4.3. Automorphisms of fusion and linking systems.......Page 161
4.4. Normal fusion and linking subsystems.......Page 164
4.5. Fundamental groups and covering spaces.......Page 167
4.6. Homotopy properties of classifying spaces.......Page 170
4.7. Classifying spectra of fusion systems.......Page 174
4.8. An infinite version: p-local compact groups.......Page 176
5. The orbit category and its applications......Page 177
5.1. Higher limits of functors and the bar resolution.......Page 179
5.2. Constrained fusion systems.......Page 184
5.3. Existence, uniqueness, and automorphisms of linking systems.......Page 191
5.4. Some computational techniques for higher limits over orbit categories.......Page 198
5.5. Homotopy colimits and homotopy decompositions.......Page 206
5.6. The subgroup decomposition of |L|.......Page 209
5.7. An outline of the proofs of Theorems 4.21 and 4.22.......Page 213
5.8. The centralizer and normalizer decompositions of |L|.......Page 216
6. Examples of exotic fusion systems......Page 218
6.1. Reduced fusion systems and tame fusion systems.......Page 219
6.2. The Ruiz-Viruel examples.......Page 221
6.3. Saturated fusion systems over 2-groups.......Page 223
6.5. Other examples.......Page 224
7. Open problems......Page 225
Part IV. Fusion and Representation theory......Page 229
1.1. Ideals and Idempotents.......Page 231
1.2. G-algebras.......Page 235
1.3. Relative trace maps and Brauer homomorphisms.......Page 236
2.1. p-permutation algebras and the Brauer homomorphisms.......Page 241
2.2. (A,G)-Brauer pairs and inclusion.......Page 244
2.3. (A, b,G)-Brauer pairs and inclusion.......Page 250
2.4. (A, b,G)-Brauer pairs and fusion systems.......Page 252
3.1. Saturated triples.......Page 253
3.2. Normaliser systems and saturated triples.......Page 258
3.3. Saturated triples and normal subgroups.......Page 260
3.4. Block fusion systems.......Page 262
3.5. Fusion systems of blocks of local subgroups.......Page 264
4. Background on finite group representations......Page 267
4.1. Ordinary and modular representations.......Page 268
4.2. p-modular systems.......Page 270
4.3. Cartan and decomposition maps.......Page 271
4.4. Ordinary and Brauer characters.......Page 275
5.1. The three main theorems of Brauer.......Page 279
5.2. Relative projectivity and representation type.......Page 283
5.3. Finiteness conjectures.......Page 285
5.4. Source algebras and Puig’s conjecture.......Page 287
5.5. Kulshammer-Puig classes.......Page 290
5.6. Nilpotent blocks and extensions.......Page 295
5.7. Counting Conjectures.......Page 297
6. Block fusion systems and normal subgroups.......Page 302
7. Open Problems......Page 307
Appendix A. Background facts about groups......Page 309
References......Page 315
List of notation......Page 323
Index......Page 326