Appendix A. Left Kan extensions preserve boundedness vii viii CONTENTS 32. Degeneracy Map 33. Bounded diagrams and left Kan extensions Appendix B. Categorical Preliminaries 34. Categories over and under an object 35. Relative version of categories over and under an object 36. Pull-back process and Kan extensions 37. Cofinality for colimits 38. Grothendieck construction 39. Grothendieck construction & the pull-back process 40. Functors indexed by Grothendieck constructions Bibliography Index
Abstract. One way to understand the mod p homotopy theory of classifying spaces of finite groups is to compute their BZ/p-cellularization. In the easiest cases this is a classifying space of a finite group (always a finite p-group). If not, weshow that it has infinitely many non-trivial homotopy groups. Moreover they are either p-torsion free or else infinitely many of them contain p-torsion. By means of techniques related to fusion systems we exhibit concrete examples where p-torsion appears.
We characterize Hopf spaces with finitely generated cohomology as an algebra over the Steenrod algebra. We "deconstruct" the original space into an H-space Y with finite mod p cohomology and a finite number of p-torsion Eilenberg-Mac Lane spaces. We give a precise description of homotopy commutative H-spaces in this setting.
The purpose of this paper is to explore the concept of localization, which comes from homotopy theory, in the context of finite simple groups. We give an easy criterion for a finite simple group to be a localization of some simple subgroup and we apply it in various cases. Iterating this process allows us to connect many simple groups by a sequence of localizations. We prove that all sporadic simple groups (except possibly the Monster) and several groups of Lie type are connected to alternating groups. The question remains open whether or not there are several connected components within the family of finite simple groups.
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