Fusion Systems in Algebra and Topology (London Mathematical Society Lecture Note Series, Vol. 391)

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. This book serves as a basic reference and as an introduction to the field, particularly for students and other young mathematicians. 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 Cover 1 LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES 2 Title 4 Copyright 5 Contents 6 Introduction 10 Part I. Introduction to fusion systems 14 1. The fusion category of a finite group 14 2. Abstract fusion systems 16 3. Alperin’s fusion theorem 20 4. Normal and central subgroups of a fusion system 26 5. Normalizer fusion systems 30 6. Normal fusion subsystems and products 34 7. Fusion subsystems of p-power index or of index prime to p 41 8. The transfer homomorphism for saturated fusion systems 47 9. Other definitions of saturation 54 Part II. The local theory of fusion systems 58 1. Notation and terminology on groups 60 2. Fusion systems 60 Exercises for Section 2 61 3. Saturated fusion systems 62 Exercises for Section 3 63 4. Models for constrained saturated fusion systems 63 Exercises for Section 4 64 5. Factor systems and surjective morphisms 65 Exercises for Section 5 69 6. Invariant subsystems of fusion systems 69 Exercises for Section 6 71 7. Normal subsystems of fusion systems 71 8. Invariant maps and normal maps 74 Exercises for Section 8 76 9. Theorems on normal subsystems 77 10. Composition series 80 Exercises for Section 10 86 11. Constrained systems 87 Exercises for Section 11 89 12. Solvable systems 89 13. Fusion systems in simple groups 93 14. Classifying simple groups and fusion systems 96 15. Systems of characteristic 2-type 102 Exercises for Section 15 111 Part III. Fusion and homotopy theory 112 1. Classifying spaces, p-completion, and the Martino-Priddy conjecture 115 1.1. Homotopy and fundamental groups. 115 1.2. CW complexes and cellular homology. 118 1.3. Classifying spaces of discrete groups. 119 1.4. The p-completion functor of Bousfield and Kan. 122 1.5. Equivalences between fusion systems of finite groups. 125 1.6. The Martino-Priddy conjecture. 126 1.7. An application: fusion in finite groups of Lie type. 127 2. The geometric realization of a category 128 2.1. Simplicial sets and their realizations. 129 2.2. The nerve of a category as a simplicial set. 131 2.3. Classifying spaces as geometric realizations of categories. 133 2.4. Fundamental groups and coverings of geometric realizations. 134 2.5. Spaces of maps. 138 3. Linking systems and classifying spaces of finite groups 142 3.1. The linking category of a finite group. 142 3.2. Fusion and linking categories of spaces. 144 3.3. Linking systems and equivalences of p-completed classifying spaces. 147 4. Abstract fusion and linking systems 148 4.1. Linking systems, centric linking systems and p-local finite groups. 149 4.2. Quasicentric subgroups and quasicentric linking systems. 152 4.3. Automorphisms of fusion and linking systems. 161 4.4. Normal fusion and linking subsystems. 164 4.5. Fundamental groups and covering spaces. 167 4.6. Homotopy properties of classifying spaces. 170 4.7. Classifying spectra of fusion systems. 174 4.8. An infinite version: p-local compact groups. 176 5. The orbit category and its applications 177 5.1. Higher limits of functors and the bar resolution. 179 5.2. Constrained fusion systems. 184 5.3. Existence, uniqueness, and automorphisms of linking systems. 191 5.4. Some computational techniques for higher limits over orbit categories. 198 5.5. Homotopy colimits and homotopy decompositions. 206 5.6. The subgroup decomposition of |L|. 209 5.7. An outline of the proofs of Theorems 4.21 and 4.22. 213 5.8. The centralizer and normalizer decompositions of |L|. 216 6. Examples of exotic fusion systems 218 6.1. Reduced fusion systems and tame fusion systems. 219 6.2. The Ruiz-Viruel examples. 221 6.3. Saturated fusion systems over 2-groups. 223 6.4. Mixing related fusion systems. 224 6.5. Other examples. 224 7. Open problems 225 Part IV. Fusion and Representation theory 229 1. Algebras and G-algebras 231 1.1. Ideals and Idempotents. 231 1.2. G-algebras. 235 1.3. Relative trace maps and Brauer homomorphisms. 236 2. p-permutation algebras, Brauer pairs and fusion systems 241 2.1. p-permutation algebras and the Brauer homomorphisms. 241 2.2. (A,G)-Brauer pairs and inclusion. 244 2.3. (A, b,G)-Brauer pairs and inclusion. 250 2.4. (A, b,G)-Brauer pairs and fusion systems. 252 3. p-permutation algebras and saturated fusion systems 253 3.1. Saturated triples. 253 3.2. Normaliser systems and saturated triples. 258 3.3. Saturated triples and normal subgroups. 260 3.4. Block fusion systems. 262 3.5. Fusion systems of blocks of local subgroups. 264 4. Background on finite group representations 267 4.1. Ordinary and modular representations. 268 4.2. p-modular systems. 270 4.3. Cartan and decomposition maps. 271 4.4. Ordinary and Brauer characters. 275 5. Fusion and structure 279 5.1. The three main theorems of Brauer. 279 5.2. Relative projectivity and representation type. 283 5.3. Finiteness conjectures. 285 5.4. Source algebras and Puig’s conjecture. 287 5.5. Kulshammer-Puig classes. 290 5.6. Nilpotent blocks and extensions. 295 5.7. Counting Conjectures. 297 6. Block fusion systems and normal subgroups. 302 7. Open Problems 307 Appendix A. Background facts about groups 309 References 315 List of notation 323 Index 326 9781107601000 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-- Machine Generated Contents Note: Introduction; 1. Introduction To Fusion Systems; 2. The Local Theory Of Fusion Systems; 3. Fusion And Homotopy Theory; 4. Fusion And Representation Theory; Appendix. Background Facts About Groups; References; List Of Notation; Index. Michael Aschbacher, Radha Kessar, Bob Oliver. Includes Index. Includes Bibliographical References (p. 306-313) And Index. "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"-- Provided by publisher This book combines a detailed exposition of the basics of fusion systems with a survey of the current state of the field