**Mathematical Surveys and Monographs**

Volume: 147;
2008;
289 pp;
Hardcover

MSC: Primary 20; 55;

**Print ISBN: 978-0-8218-4474-8
Product Code: SURV/147**

List Price: $99.00

AMS Member Price: $79.20

MAA Member Price: $89.10

**Electronic ISBN: 978-1-4704-1374-3
Product Code: SURV/147.E**

List Price: $93.00

AMS Member Price: $74.40

MAA Member Price: $83.70

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#### Supplemental Materials

# Classifying Spaces of Sporadic Groups

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*David J. Benson; Stephen D. Smith*

For each of the 26 sporadic finite simple groups, the authors construct a
2-completed classifying space using a homotopy decomposition in terms of
classifying spaces of suitable 2-local subgroups. This construction leads to
an additive decomposition of the mod 2 group cohomology. The authors also
summarize the current status of knowledge in the literature about the ring
structure of the mod 2 cohomology of sporadic simple groups.

This book begins with a fairly extensive initial exposition, intended for
non-experts, of background material on the relevant constructions from
algebraic topology, and on local geometries from group theory. The
subsequent chapters then use those structures to develop the main results on
individual sporadic groups.

#### Readership

Graduate students and research mathematicians interested in group theory and algebraic topology.

#### Reviews & Endorsements

...remarkably accessible at explaining sporadic groups and also is successful at working with the cohomology of the other simple groups.

-- SciTech Book News

The core of the book consists of a detailed discussion of each sporadic group in turn. Very usefully however, the first half of the book consists of a tailor-made introduction to the relevant areas of algebraic topology and group theory. This coherent, modern account of the wide range of topics involved by two eminent researchers and expositors means that the book should be valuable to many more people working on the boundary between algebraic topology and group theory.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Classifying Spaces of Sporadic Groups

- Contents v6 free
- Introduction ix10 free
- Some general notation and conventions xiii14 free
- Chapter 1. Overview of our main results 118 free
- Part 1. Exposition of background material 522
- Chapter 2. Review of selected aspects of group cohomology 724
- 2.1. Some features of group cohomology via the algebraic approach 724
- 2.2. Some features of group cohomology via the classifying space 926
- 2.3. The relation between the coefficient rings Z and F[sub(p)] 1229
- 2.4. The duality relation between homology and cohomology 1431
- 2.5. The Borel construction and G-equivariant homology 1633
- 2.6. Spectral sequences for G-equivariant homology 2037
- 2.7. Review of some aspects of simplicial complexes 2340

- Chapter 3. Simplicial sets and their equivalence with topological spaces 2744
- 3.1. The context of simplicial sets 2744
- 3.2. A presentation for the category Δ of finite ordered sets 2946
- 3.3. Singular simplices and geometric realization 3047
- 3.4. Products and function spaces 3249
- 3.5. Categorical settings for homotopy theory 3552
- 3.6. Model categories, Kan complexes and Quillen's equivalence 3653
- 3.7. Simplicial R-modules and homology 4057

- Chapter 4. Bousfield-Kan completions and homotopy colimits 4360
- 4.1. Cosimplicial objects 4360
- 4.2. Bousfield–Kan completions 4562
- 4.3. Completion of the classifying space BG 4865
- 4.4. Simplicial spaces 5067
- 4.5. Diagrams of spaces and homotopy colimits 5269
- 4.6. The homotopy colimit over a simplex category 6178
- 4.7. The Borel construction as a homotopy colimit 6582
- 4.8. The Bousfield–Kan homology spectral sequence 6784

- Chapter 5. Decompositions and ample collections of p-subgroups 7390
- 5.1. Some history of homology and homotopy decompositions 7491
- 5.2. Homology decompositions defined by homotopy colimits 83100
- 5.3. Ampleness for collections of subgroups 87104
- 5.4. The centralizer decomposition 89106
- 5.5. The subgroup decomposition 90107
- 5.6. The normalizer decomposition 92109
- 5.7. Ampleness for the centric collection 99116
- 5.8. Sharpness for the three decompositions 100117
- 5.9. The "standard" ample G-homotopy type defined by S[sub(p)](G) 103120
- 5.10. Ample (but potentially inequivalent) subcollections of A[sub(p)](G) 106123
- 5.11. Ample (but potentially inequivalent) subcollections of B[sub(p)](G) 107124

- Chapter 6. 2-local geometries for simple groups 109126
- 6.1. Some history of geometries for simple groups 110127
- 6.2. Preview: some initial examples of local geometries 113130
- 6.3. Some general constructions of 2-local geometries 126143
- 6.4. Flag-transitive action on geometries 136153
- 6.5. Some equivalence methods for 2-local geometries 142159
- 6.6. Final remarks on 2-local geometries 150167

- Part 2. Main results on sporadic groups 153170
- Chapter 7. Decompositions for the individual sporadic groups 155172
- 7.1. The Mathieu group M[sub(11)] 157174
- 7.2. The Mathieu group M[sub(12)] 161178
- 7.3. The Mathieu group M[sub(22)] 165182
- 7.4. The Mathieu group M[sub(23)] 172189
- 7.5. The Mathieu group M[sub(24)] 178195
- 7.6. The Janko group J[sub(1)] 180197
- 7.7. The Janko group J[sub(2)] 181198
- 7.8. The Janko group J[sub(3)] 183200
- 7.9. The Janko group J[sub(4)] 187204
- 7.10. The Higman–Sims group HS 190207
- 7.11. The McLaughlin group McL 194211
- 7.12. The Suzuki group Suz 197214
- 7.13. The Conway group Co[sub(3)] 200217
- 7.14. The Conway group Co[sub(2)] 203220
- 7.15. The Conway group Co[sub(1)] 206223
- 7.16. The Fischer group Fi[sub(22)] 208225
- 7.17. The Fischer group Fi[sub(23)] 211228
- 7.18. The Fischer group Fi'[sub(24)] 215232
- 7.19. The Harada–Norton group HN = F[sub(5)] 222239
- 7.20. The Thompson group Th = F[sub(3)] 224241
- 7.21. The Baby Monster B = F[sub(2)] 226243
- 7.22. The Fischer–Griess Monster M = F[sub(1)] 229246
- 7.23. The Held group He 231248
- 7.24. The Rudvalis group Ru 233250
- 7.25. The O'Nan group O'N 237254
- 7.26. The Lyons group Ly 242259

- Chapter 8. Details of proofs for individual groups 247264

- Bibliography 275292
- Index 281298