Classifying Spaces and Group Cohomology
This is one of the basic chapters in the book. We start in §1, §2, with preliminaries from topology. The results reviewed in §1 on the basic structure and properties of classifying spaces are essential throughout the remainder of the text. The material in §2 on the Steenrod algebra is not needed in the rest of this chapter and is placed here only for continuity. It is used, however, in Chap. III, §3, and, from then on, more and more frequently throughout the book. Here we only review the basic facts and give Steenrod’s axiomatic treatment of the p th power operations. However, in Chap. IV, §7 we will use the cohomology of groups to provide the construction of the Steenrod operations. In §3 we give the definition of group cohomology and devote the remainder of this chapter, with the exception of §8 to basic facts and techniques for calculating these cohomology groups, particularly for finite groups. §8 gives a nice application of these ideas to construct non-trivial outer automorphisms for p-groups.
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