Introduction Aims of the book Much of the material in this book originated as a set of lecture notes for a graduate course titled Subgroup Complexes, given in Fall 1990 at Notre Dame, and in Fall 1994 at the University of Illinois at Chicago (UIC) a later draft was used for the summer graduate seminar at UIC during Summer 2005. This final version is also intended—at least in part—for possible use as a grad- uate text for such a course especially in the chapters in the first half or so, where usually reasonably full details are given. The level of the course would typically be second-year: the exposition should ideally be accessible to students who have taken first-year graduate algebra—but who may have seen at most some basics of algebraic topology and combinatorics. The choice of the material aims to give a rough overview of a modern research area, from its historical origins through some directions of current research problems. The treatment includes proofs of many basic and intermediate results, as well as examples and exercises. On the other hand, especially in the later chapters, more typically proofs are just sketched, or omitted and fuller references are given, which ideally the more motivated reader can follow up. Thus in these areas, the material is perhaps more appropriate for an advanced reading course. Finally, the book is also intended to be of at least some use as a reasonably general reference for this research area: So as indicated above I have also included statements of various more advanced results typically without proof, since the proofs would involve details beyond the fairly elementary setting of the book. In selecting the topics, I have made an effort to collect and compare material from various sources (some of it apparently in the “folklore” category) and to cover various areas reasonably well known to me. Of course I cannot hope to be truly encyclopedic about the many research directions related to subgroup complexes—so I apologize in advance for omissions of any favorite material of potential readers. (In particular because of my various other commitments after this book was started, in various areas I have fallen behind developments that have taken place since about 1994 so I thank numerous colleagues who have supplied more up-to-date references in many of these areas.) Optional tracks (B,S,G) in reading the book An additional feature of the treatment is that I have included many examples of subgroup complexes related to finite simple groups, which is my own particular area of interest. It seemed sensible to present this material as “optional” reading— since the details of the structure of all but the smallest simple groups may not be 1
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