Math 283
Winter 2012

Home Syllabus

This schedule is only tentative, and may be adjusted as necessary. In the references column, you will find references for that day's lecture, as well as other materials that were mentioned.

Date Topics References
Week 1
Jan 09 Historical motivation and applications.
  • Cohomology of groups, by K.S. Brown. (Introduction)
  • Der Cohomologie Ring einer beliebigen Gruppe, by B. Eckmann.
  • Relations between homology and homotopy groups I, II, by S. Eilenberg and S. MacLane.
  • Der Einfluss der Fundamentalgruppe auf die Bettischen Gruppen, by H. Freudental.
  • Fundamentalgruppe und zweite Bettische Gruppe, by H. Hopf.
  • Selecta Heinz Hopf, by H. Hopf.
  • Über die Bettische Gruppen, die zu einer beliebige Gruppe gehören, by H. Hopf.
Jan 10
Jan 11 Group homology from the point of view of homological algebra. Cohomology of groups, by K.S. Brown. (I.0-3, I.7-8, II.1-3, III.0-1)
Jan 12
Jan 13 The relationship between group (co)homology and the functors Tor, Ext. Cohomology of groups, by K.S. Brown. (III.0-2)
Week 2
Jan 16 Martin Luther King Jr. Day. No class today.
Jan 17
Jan 18 The bar resolution and the classifying space.
  • Cohomology of groups, by K.S. Brown (I.5, II.4, II.6, III.6, III.8)
  • Cohomology of finite groups, by A. Adem and R.J. Milgram. (II.1, II.3)
  • The cohomology of groups, by L. Evens (2.3)
  • Representations and cohomology I, by D.J. Benson (3.4)
  • Representations and cohomology II, by D.J. Benson (2.1-2.4)
Jan 19
Jan 20 Milnor's construction and the simplicial construction of the classifying space.
  • Cohomology of finite groups, by A. Adem and R.J. Milgram. (II.1)
  • Representations and cohomology II, by D.J. Benson (2.3-2.4)
Week 3
Jan 23 No class today. To be rescheduled.
Jan 24
Jan 25 No class today. To be rescheduled.
Jan 26
Jan 27 Functoriality, restriction and transfer maps.
  • Cohomology of groups, by K.S. Brown (III.6, III.8, III.9-10)
  • Cohomology of finite groups, by A. Adem and R.J. Milgram. (II.5)
  • The cohomology of groups, by L. Evens (4.1, 4.2)
  • Representations and cohomology II, by D.J. Benson (2.7)
Week 4
Jan 30
Jan 31
Feb 01 Transfer and the primary decomposition
  • Cohomology of groups, by K.S. Brown (III.9-10)
  • Cohomology of finite groups, by A. Adem and R.J. Milgram. (II.5)
  • The cohomology of groups, by L. Evens (4.1, 4.2)
  • Representations and cohomology II, by D.J. Benson (2.7)
Feb 02
Feb 03 Invariants and stable elements
  • Cohomology of groups, by K.S. Brown (III.10)
  • Cohomology of finite groups, by A. Adem and R.J. Milgram. (II.6)
  • The cohomology of groups, by L. Evens (4.2)
Week 5
Feb 06
Feb 07
Feb 08 Fibrations and spectral sequences
  • Cohomology of groups, by K.S. Brown. (VII.1-6)
  • A user's guide to spectral sequences, by J. McCleary
  • Spectral sequences in algebraic topology, by A. Hatcher (Chapter 1)
  • Representations and cohomology II, by D.J. Benson (3.1-3.5)
Feb 09
Feb 10 Spectral sequence computations
  • Cohomology of groups, by K.S. Brown. (VII.1-6)
  • A user's guide to spectral sequences, by J. McCleary
  • Spectral sequences in algebraic topology, by A. Hatcher (Chapter 1)
  • Representations and cohomology II, by D.J. Benson (3.9)
Week 6
Feb 13
Feb 14
Feb 15 The second (co)homology group and extensions. Cohomology of groups, by K.S. Brown. (I.5, IV.3)
Feb 16
Feb 17 The cohomology of finite categories and higher limits. Representations and cohomology of finite categories, by F. Xu. (1.1.3, 3.1)
Week 7
Feb 20 Presidents' day. No class today.
Feb 21
Feb 22 The bar resolution for finite categories. Representations and cohomology of finite categories, by F. Xu. (1.2.2, 2, 4.1.2)
Feb 23
Feb 24 Homotopy colimits and obstructions I.
Week 8
Feb 27 Homotopy colimits and obstructions II.
Feb 28
Feb 29 Quillen's solution to the Atiyah-Swan conjecture I. The spectrum of an equivariant cohomology ring I and II, by D. Quillen.
Mar 01
Mar 02 Quillen's solution to the Atiyah-Swan conjecture II. The spectrum of an equivariant cohomology ring I and II, by D. Quillen.
Week 9
Mar 05
Mar 06
Mar 07 Periodic cohomology and free actions on spheres.
  • Cohomology of finite groups, by A. Adem and R.J. Milgram. (IV.6)
  • Groups which act on Sn without fixed point, by J. Milnor.
Mar 08
Mar 09 Milnor's condition and free actions on homotopy spheres.
  • Groups which act on Sn without fixed point, by J. Milnor.
  • Cohomology of groups, by K.S. Brown (I.6)
  • Periodic resolutions for finite groups, by R.G. Swan.
Week 10
Mar 12
Mar 13
Mar 14 Periodic resolutions and homotopy spheres.
  • Periodic resolutions for finite groups, by R.G. Swan.
  • Induced representations and projective modules, by R.G. Swan.
Mar 15
Mar 16 The spherical space form problem and free actions of groups on products of spheres.
  • Periodic resolutions for finite groups, by R.G. Swan.
  • Induced representations and projective modules, by R.G. Swan.
  • Constructing and deconstructing group actions, by A. Adem.

Winter 2012-- Department of Mathematics, Stanford University
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