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Fusion Systems in Algebra and Topology [Pehme köide]

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(California Institute of Technology), (University of Aberdeen), (Université de Paris XIII)
  • Formaat: Paperback / softback, 330 pages, kõrgus x laius x paksus: 226x150x18 mm, kaal: 490 g, Worked examples or Exercises; 30 Tables, unspecified; 30 Plates, unspecified
  • Sari: London Mathematical Society Lecture Note Series
  • Ilmumisaeg: 25-Aug-2011
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1107601002
  • ISBN-13: 9781107601000
Teised raamatud teemal:
Fusion Systems in Algebra and TopologySuurem pilt
  • Formaat: Paperback / softback, 330 pages, kõrgus x laius x paksus: 226x150x18 mm, kaal: 490 g, Worked examples or Exercises; 30 Tables, unspecified; 30 Plates, unspecified
  • Sari: London Mathematical Society Lecture Note Series
  • Ilmumisaeg: 25-Aug-2011
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1107601002
  • ISBN-13: 9781107601000
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781107601000.html
Märksõnad:
"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.

Muu info

This book combines a detailed exposition of the basics of fusion systems with a survey of the current state of the field.
Introduction 1(4)
Part I Introduction to fusion systems
5(44)
1 The fusion category of a finite group
5(2)
2 Abstract fusion systems
7(4)
3 Alperin's fusion theorem
11(6)
4 Normal and central subgroups of a fusion system
17(4)
5 Normalizer fusion systems
21(4)
6 Normal fusion subsystems and products
25(7)
7 Fusion subsystems of p-power index or of index prime to p
32(6)
8 The transfer homomorphism for saturated fusion systems
38(7)
9 Other definitions of saturation
45(4)
Part II The local theory of fusion systems
49(54)
1 Notation and terminology on groups
51(1)
2 Fusion systems
51(2)
3 Saturated fusion systems
53(1)
4 Models for constrained saturated fusion systems
54(2)
5 Factor systems and surjective morphisms
56(4)
6 Invariant subsystems of fusion systems
60(2)
7 Normal subsystems of fusion systems
62(3)
8 Invariant maps and normal maps
65(3)
9 Theorems on normal subsystems
68(3)
10 Composition series
71(7)
11 Constrained systems
78(2)
12 Solvable systems
80(4)
13 Fusion systems in simple groups
84(3)
14 Classifying simple groups and fusion systems
87(6)
15 Systems of characteristic 2-type
93(10)
Part III Fusion and homotopy theory
103(117)
1 Classifying spaces, p-completion, and the Martino-Priddy conjecture
106(13)
1.1 Homotopy and fundamental groups
106(3)
1.2 CW complexes and cellular homology
109(1)
1.3 Classifying spaces of discrete groups
110(3)
1.4 The p-completion functor of Bousfield and Kan
113(3)
1.5 Equivalences between fusion systems of finite groups
116(1)
1.6 The Martino-Priddy conjecture
117(1)
1.7 An application: fusion in finite groups of Lie type
118(1)
2 The geometric realization of a category
119(14)
2.1 Simplicial sets and their realizations
120(2)
2.2 The nerve of a category as a simplicial set
122(2)
2.3 Classifying spaces as geometric realizations of categories
124(1)
2.4 Fundamental groups and coverings of geometric realizations
125(4)
2.5 Spaces of maps
129(4)
3 Linking systems and classifying spaces of finite groups
133(6)
3.1 The linking category of a finite group
133(2)
3.2 Fusion and linking categories of spaces
135(3)
3.3 Linking systems and equivalences of p-completed classifying spaces
138(1)
4 Abstract fusion and linking systems
139(29)
4.1 Linking systems, centric linking systems and p-local finite groups
140(3)
4.2 Quasicentric subgroups and quasicentric linking systems
143(9)
4.3 Automorphisms of fusion and linking systems
152(3)
4.4 Normal fusion and linking subsystems
155(3)
4.5 Fundamental groups and covering spaces
158(3)
4.6 Homotopy properties of classifying spaces
161(4)
4.7 Classifying spectra of fusion systems
165(2)
4.8 An infinite version: p-local compact groups
167(1)
5 The orbit category and its applications
168(41)
5.1 Higher limits of functors and the bar resolution
170(5)
5.2 Constrained fusion systems
175(7)
5.3 Existence, uniqueness, and automorphisms of linking systems
182(7)
5.4 Some computational techniques for higher limits over orbit categories
189(8)
5.5 Homotopy colimits and homotopy decompositions
197(3)
5.6 The subgroup decomposition of |L|
200(4)
5.7 An outline of the proofs of Theorems 4.21 and 4.22
204(3)
5.8 The centralizer and normalizer decompositions of |L|
207(2)
6 Examples of exotic fusion systems
209(7)
6.1 Reduced fusion systems and tame fusion systems
210(2)
6.2 The Ruiz-Viruel examples
212(2)
6.3 Saturated fusion systems over 2-groups
214(1)
6.4 Mixing related fusion systems
215(1)
6.5 Other examples
215(1)
7 Open problems
216(4)
Part IV Fusion and Representation theory
220(80)
1 Algebras and G-algebras
222(10)
1.1 Ideals and Idempotents
222(4)
1.2 G-algebras
226(1)
1.3 Relative trace maps and Brauer homomorphisms
227(5)
2 p-permutation algebras, Brauer pairs and fusion systems
232(12)
2.1 p-permutation algebras and the Brauer homomorphisms
232(3)
2.2 (A, G)-Brauer pairs and inclusion
235(6)
2.3 (A, b, G)-Brauer pairs and inclusion
241(2)
2.4 (A, b, G)-Brauer pairs and fusion systems
243(1)
3 p-permutation algebras and saturated fusion systems
244(14)
3.1 Saturated triples
244(5)
3.2 Normaliser systems and saturated triples
249(2)
3.3 Saturated triples and normal subgroups
251(2)
3.4 Block fusion systems
253(2)
3.5 Fusion systems of blocks of local subgroups
255(3)
4 Background on finite group representations
258(12)
4.1 Ordinary and modular representations
259(2)
4.2 p-modular systems
261(1)
4.3 Cartan and decomposition maps
262(4)
4.4 Ordinary and Brauer characters
266(4)
5 Fusion and structure
270(23)
5.1 The three main theorems of Brauer
270(4)
5.2 Relative projectivity and representation type
274(2)
5.3 Finiteness conjectures
276(2)
5.4 Source algebras and Puig's conjecture
278(3)
5.5 Kulshammer-Puig classes
281(5)
5.6 Nilpotent blocks and extensions
286(2)
5.7 Counting Conjectures
288(5)
6 Block fusion systems and normal subgroups
293(5)
7 Open Problems
298(2)
Appendix A Background facts about groups 300(6)
References 306(8)
List of notation 314(3)
Index 317
Michael Aschbacher is the Shaler Arthur Hanisch Professor of Mathematics at the California Institute of Technology. Radha Kessar is a Reader in the Institute of Mathematics at the University of Aberdeen. Bob Oliver is a Professor in the Laboratoire Analyse, Géométrie et Applications (LAGA) at the Université de Paris XIII.