# Group Cohomology and Algebraic Cycles

Update (January 11, 2014): Now proofreading. I’ve found a typo in the title of Chapter 8. I misspelled ‘ring’ — pretty humbling. (To give due credit, my editor spotted the typo.)

Original (December 4, 2013):
More about the book at the CUP site (it’s not available yet — I’m checking the copyedited manuscript right now).

You can read the preface in manuscript here.

Contents
Preface
1 Group cohomology
1.1 Definition of group cohomology
1.2 Equivariant cohomology and basic calculations
1.3 Algebraic definition of group cohomology
2 The Chow ring of a classifying space
2.1 The Chow group of algebraic cycles
2.2 The Chow ring of a classifying space
2.3 The equivariant Chow ring
2.4 Basic computations
2.5 Transfer
2.6 Becker–Gottlieb transfer for Chow groups
2.7 Groups in characteristic
2.8 Wreath products and the symmetric groups
2.9 General linear groups over finite fields
2.10 Questions about the Chow ring of a finite group
3 Depth and regularity
3.1 Depth and regularity in terms of local cohomology
3.2 Depth and regularity in terms of generators and relations
3.3 Duflot’s lower bound for depth
4 Regularity of group cohomology
4.1 Regularity of group cohomology and applications
4.2 Proof of Symonds’s theorem
5 Generators for the Chow ring
5.1 Bounding the generators of the Chow ring
5.2 Optimality of the bounds
6 Regularity of the Chow ring
6.1 Bounding the regularity of the Chow ring
6.2 Motivic cohomology
6.3 Steenrod operations on motivic cohomology
6.4 Regularity of motivic cohomology
7 Bounds for p-groups
7.1 Invariant theory of the group Z = p
7.2 Wreath products
7.3 Bounds for the Chow ring and cohomology of a p-group
8 The structure of group cohomology and the Chow ring
8.1 The norm map
8.2 Quillen’s theorem and Yagita’s theorem
8.3 Yagita’s theorem over any field
8.4 Carlson’s theorem on transfer
9 Cohomology mod transfers is Cohen–Macaulay
9.1 The Cohen-Macaulay property
9.2 The ring of invariants modulo traces
10 Bounds for group cohomology and the Chow ring modulo transfers
11 Transferred Euler classes
11.1 Basic properties of transferred Euler classes
11.2 Generating the Chow ring
12 Detection theorems for cohomology and Chow rings
12.1 Nilpotence in group cohomology
12.2 The detection theorem for Chow rings
13 Calculations
13.1 The Chow rings of the groups of order 16
13.2 The modular p-group
13.3 Central extensions by Gm
13.4 The extraspecial group Ep3
13.5 Calculations of the topological nilpotence degree
14 Groups of order p4
14.1 The wreath product Z/3 ≀ Z/3
14.2 Geometric and topological filtrations
14.3 Groups of order p4 for p ≥ 5
14.4 Groups of order 81
14.5 A 1-dimensional group
15 Geometric and topological filtrations
15.1 Summary
15.2 Positive results
15.3 Examples at odd primes
15.4 Examples for p = 2
16 The Eilenberg–Moore spectral sequence in motivic cohomology
16.1 Motivic cohomology of flag bundles
16.2 Leray spectral sequence for a divisor with normal crossings
16.3 Eilenberg-Moore spectral sequence in motivic cohomology
17 The Chow K¨unneth conjecture
18 Open problems
Appendix Tables
References
Index