Classifying spaces of sporadic groups / David J. Benson, Stephen D. Smith.

Format:

Book

Author/Creator:

Benson, D. J. (David J.), 1955- author.

Smith, Stephen D., 1948- author.

Series:

Mathematical surveys and monographs ; no. 147.

Mathematical surveys and monographs, 0076-5376 ; volume 147

Language:

English

Subjects (All):

Sporadic groups (Mathematics).

Classifying spaces.

Physical Description:

1 online resource (310 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2008]

Language Note:

English

Summary:

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.

Contents:

""Contents""; ""Introduction""; ""Some general notation and conventions""; ""Chapter 1. Overview of our main results""; ""Part 1. Exposition of background material""; ""Chapter 2. Review of selected aspects of group cohomology""; ""2.1. Some features of group cohomology via the algebraic approach""; ""2.2. Some features of group cohomology via the classifying space""; ""2.3. The relation between the coefficient rings Z and F[sub(p)]""; ""2.4. The duality relation between homology and cohomology""; ""2.5. The Borel construction and G-equivariant homology""

""2.6. Spectral sequences for G-equivariant homology""""2.7. Review of some aspects of simplicial complexes""; ""Chapter 3. Simplicial sets and their equivalence with topological spaces""; ""3.1. The context of simplicial sets""; ""3.2. A presentation for the category Δ of finite ordered sets""; ""3.3. Singular simplices and geometric realization""; ""3.4. Products and function spaces""; ""3.5. Categorical settings for homotopy theory""; ""3.6. Model categories, Kan complexes and Quillen's equivalence""; ""3.7. Simplicial R-modules and homology""

""Chapter 4. Bousfield-Kan completions and homotopy colimits""""4.1. Cosimplicial objects""; ""4.2. Bousfield�Kan completions""; ""4.3. Completion of the classifying space BG""; ""4.4. Simplicial spaces""; ""4.5. Diagrams of spaces and homotopy colimits""; ""4.6. The homotopy colimit over a simplex category""; ""4.7. The Borel construction as a homotopy colimit""; ""4.8. The Bousfield�Kan homology spectral sequence""; ""Chapter 5. Decompositions and ample collections of p-subgroups""; ""5.1. Some history of homology and homotopy decompositions""

""5.2. Homology decompositions defined by homotopy colimits""""5.3. Ampleness for collections of subgroups""; ""5.4. The centralizer decomposition""; ""5.5. The subgroup decomposition""; ""5.6. The normalizer decomposition""; ""5.7. Ampleness for the centric collection""; ""5.8. Sharpness for the three decompositions""; ""5.9. The ""standard"" ample G-homotopy type defined by S[sub(p)](G)""; ""5.10. Ample (but potentially inequivalent) subcollections of A[sub(p)](G)""; ""5.11. Ample (but potentially inequivalent) subcollections of B[sub(p)](G)""

""Chapter 6. 2-local geometries for simple groups""""6.1. Some history of geometries for simple groups""; ""6.2. Preview: some initial examples of local geometries""; ""6.3. Some general constructions of 2-local geometries""; ""6.4. Flag-transitive action on geometries""; ""6.5. Some equivalence methods for 2-local geometries""; ""6.6. Final remarks on 2-local geometries""; ""Part 2. Main results on sporadic groups""; ""Chapter 7. Decompositions for the individual sporadic groups""; ""7.1. The Mathieu group M[sub(11)]""; ""7.2. The Mathieu group M[sub(12)]""

""7.3. The Mathieu group M[sub(22)]""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 275-279) and index.

Description based on print version record.

ISBN:

1-4704-1374-4