Contents Preface and Acknowledgments xi Introduction 1 Aims of the book 1 Optional tracks (B,S,G) in reading the book 1 A preview via some history of subgroup complexes 2 Part 1. Background Material and Examples 7 Chapter 1. Background: Posets, simplicial complexes, and topology 9 1.1. Subgroup posets 10 1.2. Subgroup complexes 17 1.3. Topology for subgroup posets and complexes 23 1.4. Mappings for posets, complexes, and spaces 26 1.5. Group actions on posets, complexes, and spaces 28 1.6. Some further constructions related to complexes 31 Chapter 2. Examples: Subgroup complexes as geometries for simple groups 39 Introduction: Finite simple groups and their “natural” geometries 40 2.1. Motivating cases: Projective geometries for matrix groups 45 2.2. (Option B): The model case: Buildings for Lie type groups 59 Exhibiting the building via parabolic subgroups 61 Associating the Dynkin diagram to the geometry of the building 75 2.3. (Option S): Diagram geometries for sporadic simple groups 82 A general setting for geometries with associated diagrams 82 Some explicit examples of sporadic geometries 86 Part 2. Fundamental Techniques 101 Chapter 3. Contractibility 103 Preview: Cones and contractibility in subgroup posets 104 3.1. Topological background: Homotopy of maps, and homotopy equivalence of spaces 104 3.2. Cones (one-step contractibility) 111 3.3. Conical (two-step) contractibility 116 3.4. Multi-step contractibility and collapsibility 127 3.5. (Option G): G-homotopy equivalence and G-contractibility 137 Chapter 4. Homotopy equivalence 141 4.1. Topological background: Homotopy via a contractible carrier 141 4.2. Equivalences via Quillen’s Fiber Theorem 147 vii
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