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. 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 Cover; Title; Copyright; Contents; Introduction; Part I. Introduction to fusion systems; 1. The fusion category of a finite group; 2. Abstract fusion systems; 3. Alperin's fusion theorem; 4. Normal and central subgroups of a fusion system; 5. Normalizer fusion systems; 6. Normal fusion subsystems and products; 7. Fusion subsystems of p-power index or of index prime to p; 8. The transfer homomorphism for saturated fusion systems; 9. Other definitions of saturation; Part II. The local theory of fusion systems; 1. Notation and terminology on groups; 2. Fusion systems; Exercises for Section 2 14. Classifying simple groups and fusion systems15. Systems of characteristic 2-type; Exercises for Section 15; Part III. Fusion and homotopy theory; 1. Classifying spaces, p-completion, and the Martino-Priddy conjecture; 1.1. Homotopy and fundamental groups; 1.2. CW complexes and cellular homology; 1.3. Classifying spaces of discrete groups; 1.4. The p-completion functor of Bousfield and Kan; 1.5. Equivalences between fusion systems of finite groups; 1.6. The Martino-Priddy conjecture; 1.7. An application: fusion in finite groups of Lie type; 2. The geometric realization of a category 2.1. Simplicial sets and their realizations2.2. The nerve of a category as a simplicial set; 2.3. Classifying spaces as geometric realizations of categories; 2.4. Fundamental groups and coverings of geometric realizations; 2.5. Spaces of maps; 3. Linking systems and classifying spaces of finite groups; 3.1. The linking category of a finite group; 3.2. Fusion and linking categories of spaces; 3.3. Linking systems and equivalences of p-completed spaces; 4. Abstract fusion and linking systems; 4.1. Linking systems, centric linking systems and p-local groups 3. Saturated fusion systemsExercises for Section 3; 4. Models for constrained saturated fusion systems; Exercises for Section 4; 5. Factor systems and surjective morphisms; Exercises for Section 5; 6. Invariant subsystems of fusion systems; Exercises for Section 6; 7. Normal subsystems of fusion systems; 8. Invariant maps and normal maps; Exercises for Section 8; 9. Theorems on normal subsystems; 10. Composition series; Exercises for Section 10; 11. Constrained systems; Exercises for Section 11; 12. Solvable systems; 13. Fusion systems in simple groups 4.2. Quasicentric subgroups and quasicentric linking4.3. Automorphisms of fusion and linking systems; 4.4. Normal fusion and linking subsystems; 4.5. Fundamental groups and covering spaces; 4.6. Homotopy properties of classifying spaces; 4.7. Classifying spectra of fusion systems; 4.8. An infinite version: p-local compact groups; 5. The orbit category and its applications; 5.1. Higher limits of functors and the bar resolution; 5.2. Constrained fusion systems; 5.3. Existence, uniqueness, and automorphisms of linking systems This book combines a detailed exposition of the basics of fusion systems with a survey of the current state of the field 5.4. Some computational techniques for higher limits over orbit categories