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http://dx.doi.org/10.1090/surv/147 Mathematical I Surveys I and I Monographs I Volume 147 I Classifying Spaces of Sporadic Groups I David J. Benson I Stephen D. Smith Editorial Committee Jerry L. Bona Michael G. Eastwood Ralph L. Cohen Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 20D08, 20J06, 55R35; Secondary 20B25, 20J05, 55R40. For additional information and updates on this book, visit www.ams.org/bookpages/surv-147 Library of Congress Cataloging-in-Publication Data Benson, D. J. (David J.), 1955- Classifying spaces of sporadic groups / David J. Benson, Stephen D. Smith, p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 147) Includes bibliographical references and index. ISBN 978-0-8218-4474-8 (alk. paper) 1. Sporadic groups (Mathematics) 2. Classifying spaces. I. Smith, Stephen D., 1948- II. Title. QA177.B465 2008 512'.2—dc22 2007060587 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2008 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: //www. ams. org/ 10 9 8 7 6 5 4 3 2 1 13 12 11 10 09 08 ABSTRACT. For each of the 26 sporadic finite simple groups, we construct a 2-completed classifying space, via a homotopy decomposition in terms of classifying spaces of suitable 2-local subgroups; this leads to an additive de­ composition of the mod 2 group cohomology. We also summarize the current status of knowledge in the literature about the ring structure of the mod 2 cohomology of those groups. Our decompositions arise via recent homotopy colimit theorems of various authors: in those results, the colimit is indexed by a collection of 2-subgroups, which is "ample" in the sense of affording the desired cohomology decom­ position. Furthermore our decompositions are "sharp" in the sense of being simplified by the collapse of an underlying spectral sequence. Among the various standard ample collections available in the topological literature, we make a suitably minimal choice for each group—and we further interpret that collection in terms of an equivalent "2-local geometry" from the group theory literature: namely as a simplicial complex determined by certain 2-local subgroups. In particular, we complete the verification that for each sporadic group, an appropriate 2-local geometry affords a small ample collection. One feature which emerges in each case is that the geometry has "flag-transitive" action by the group, so that the orbit complex (the quotient space modulo that action) is a simplex: and then the diagram of classifying spaces of subgroups which indexes the homotopy decomposition is a pushout n- cube of the relevant dimension n (though sometimes that orbit simplex diagram is further simplified via cancellations). The work begins with a fairly extensive initial exposition, intended for non-experts, of background material on the relevant constructions from alge­ braic topology, and on local geometries from group theory. The subsequent chapters then use those structures to develop the main results on individual sporadic groups. Contents Introduction ix Some general notation and conventions xiii Chapter 1. Overview of our main results 1 Part 1. Exposition of background material 5 Chapter 2. Review of selected aspects of group cohomology 7 2.1. Some features of group cohomology via the algebraic approach 7 2.2. Some features of group cohomology via the classifying space 9 2.3. The relation between the coefficient rings Z and ¥p 12 2.4. The duality relation between homology and cohomology 14 2.5. The Borel construction and G-equi variant homology 16 2.6. Spectral sequences for G-equi variant homology 20 2.7. Review of some aspects of simplicial complexes 23 Chapter 3. Simplicial sets and their equivalence with topological spaces 27 3.1. The context of simplicial sets 27 3.2. A presentation for the category A of finite ordered sets 29 3.3. Singular simplices and geometric realization 30 3.4. Products and function spaces 32 3.5. Categorical settings for homotopy theory 35 3.6. Model categories, Kan complexes and Quillen's equivalence 36 3.7. Simplicial R-modules and homology 40 Chapter 4. Bousfield-Kan completions and homotopy colimits 43 4T. Cosimplicial objects 43 4.2. Bousfield-Kan completions 45 4.3. Completion of the classifying space BG 48 4.4. Simplicial spaces 50 4.5. Diagrams of spaces and homotopy colimits 52 4.6. The homotopy colimit over a simplex category 61 4.7. The Borel construction as a homotopy colimit 65 4.8. The Bousfield-Kan homology spectral sequence 67 Chapter 5. Decompositions and ample collections of p-subgroups 73 5.1. Some history of homology and homotopy decompositions 74 5.2. Homology decompositions defined by homotopy colimits 83 5.3. Ampleness for collections of subgroups 87 5.4. The centralizer decomposition 89 vi CONTENTS 5.5. The subgroup decomposition 90 5.6. The normalizer decomposition 92 5.7. Ampleness for the centric collection 99 5.8. Sharpness for the three decompositions 100 5.9. The "standard" ample G-homotopy type defined by SP(G) 103 5.10. Ample (but potentially inequivalent) subcollections of AP(G) 106 5.11. Ample (but potentially inequivalent) subcollections of BP(G) 107 Chapter 6. 2-local geometries for simple groups 109 6.1. Some history of geometries for simple groups 110 6.2. Preview: some initial examples of local geometries 113 6.3. Some general constructions of 2-local geometries 126 6.4. Flag-transitive action on geometries 136 6.5. Some equivalence methods for 2-local geometries 142 6.6. Final remarks on 2-local geometries 150 Part 2. Main results on sporadic groups 153 Chapter 7. Decompositions for the individual sporadic groups 155 7.1. The Mathieu group Mn 157 7.2. The Mathieu group M12 161 7.3. The Mathieu group M22 165 7.4. The Mathieu group M23 172 7.5. The Mathieu group M24 178 7.6. The Janko group Jx 180 7.7. The Janko group J2 181 7.8. The Janko group J3 183 7.9. The Janko group J4 187 7.10. The Higman-Sims group HS 190 7.11. The McLaughlin group McL 194 7.12. The Suzuki group Suz 197 7.13. The Conway group Co3 200 7.14. The Conway group Co2 203 7.15. The Conway group Co1 206 7.16. The Fischer group Fi22 208 7.17. The Fischer group Fi23 211 7.18. The Fischer group Fi'24 215 7.19. The Harada-Norton group HN = F5 222 7.20. The Thompson group Th = F3 224 7.21. The Baby Monster B = F2 226 7.22. The Fischer-Griess Monster M = F1 229 7.23. The Held group He 231 7.24. The Rudvalis group Ru 233 7.25. The O'Nan group O 'N 237 7.26. The Lyons group Ly 242 Chapter 8. Details of proofs for individual groups 247 8.1. Details for M12 247 8.2. Details for J3 249 CONTENTS 8.3. Details for HS 250 8.4. Details for Fi22 253 8.5. Details for Fz'24 255 8.6. Details for HN 264 8.7. Details for Ru 265 8.8. Details for O'N 272 Bibliography 275 Index 281 Introduction The 26 sporadic simple groups have played an exceptional role in the theory and classification1 of the finite simple groups, ever since the discovery of the first few such groups in the late 1800s. In the last decade or so, various authors have also observed exceptional behavior in the mod p group cohomology of some of the sporadic groups, especially at the prime p — 2; see for example Benson [24, 25, 26], Benson and Wilkerson [29], Milgram [92, 93, 94]. Determining (or at least describ­ ing) the cohomology of sporadic groups represents an area of considerable current activity; and such study combines methods of algebraic topology with information and techniques from finite group theory. In this work, we provide for each sporadic group a description of a 2-completed classifying space, in terms of classifying spaces of suitable 2-local subgroups. In particular, this leads to an additive decomposition of the mod 2 group cohomology. Similar decompositions can be obtained for odd primes p; but the indexing set of p-local subgroups is usually smaller and often less interesting. Although our focus here is on sporadic groups, we emphasize that this work should be also be viewed in relation to work on cohomology of the other simple groups. For example, our decompositions for sporadic groups are largely inspired by a standard decomposition (see Example 5.1.18) of the mod p cohomology for a group of Lie type of characteristic p. However, even in that model case for Lie type groups, it remains a major open problem to give an explicit description of the individual component terms and their cohomology.
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