Classifying Spaces of Sporadic Groups (Mathematical Surveys and Monographs) by David J. Benson (PDF)

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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.

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