Classifying Spaces and Group Cohomology

  • Alejandro Adem
  • R. James Milgram
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 309)


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.


Exact Sequence Hopf Algebra Group Cohomology Double Coset Projective Resolution 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Alejandro Adem
    • 1
  • R. James Milgram
    • 2
  1. 1.Department of Mathematics, Van Vleck HallUniversity of WisconsinMadisonUSA
  2. 2.Department of MathematicsStanford UniversityStanfordUSA

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