Cohomology operations and the Steenrod algebra
Graduate Student Learning Seminar
Time Friday 12:00pm - 1:00pm & 2:00pm - 3:00pm
Place Peter Hall, Seminar Room 107
Dr Xing Gu
Dr Arik Wilbert
Dr Huijun Yang

The action of cohomology operations provides useful additional structure on the cohomology ring of a space. For a prime number p, the algebra of all stable cohomology operations on cohomology with Z/p-coefficients is known as the Steenrod algebra and admits an explicit characterization in terms of generators and relations. We will construct the Steenrod algebra for p=2, verify its properties and give some applications.

We will follow the textbook Cohomology operations and applications in homotopy theory by R. E. Mosher and M. C. Tangora [MT68]. Most of the talks will be given by the organizers. However, talks 6, 12 and 13, which consist of applications and computations, are given by students. The seminar schedule is based on a former seminar organized by T. Macko and S. Sagave at the University of Bonn.

Schedule

Talk 1, 16/02/2018 (Arik): Overview Talk (Lecture Notes)
Talk 2, 23/02/2018 (Xing): Computation: The cohomology ring of K(Z/2,1) (Lecture Notes)
[MT68, Chapter 2, p. 12-14]
Talk 3, 02/03/2018 (Xing/Huijun): Construction of the cup-i products and Steenrod squares (Lecture Notes)
[MT68, Chapter 2, p. 15-21] and [Bre97, VI §16]
Talk 4, 09/03/2018 (Huijun): Properties of the Steenrod squares (Lecture Notes)
[MT68, Chapter 3, p. 22-28]
Talk 5, 16/03/2018 (Arik): The Adem relations (Lecture Notes)
[MT68, Chapter 3, p. 29-31] and [BM82]
Talk 6, 23/03/2018 (Michelle): Application: The Hopf invariant
[MT68, Chapter 4, p. 33-38]
Talk 7, 13/04/2018 (Arik): The mod-2 Steenrod algebra (Lecture Notes)
[MT68, Chapter 5, p. 45-50]
Talk 8, 20/04/2018 (Arik): The dual of the mod-2 Steenrod algebra
[MT68, Chapter 5, p. 50-57] and [Mil58]
Talk 9, 04/05/2018 (Xing): The Serre spectral sequence (Lecture Notes)
[Hat, p. 1-13] and [MC01]
Talk 10, 11/05/2018 (Xing): Transgression and the cohomology spectral sequence of a fibration (Lecture Notes)
[MT68, Chapter 8, p. 80-81] and [MC01]
Talk 11, 18/05/2018 (Matt): Computation: The cohomology ring of K(Z/2,2)
[MT68, Chapter 9, p. 83-88]
Talk 12, 25/05/2018 (Csaba): Computation: The cohomology ring of K(Z/2,q)
[MT68, Chapter 9, p. 88-92]

References

[BM82] S. R. Bullet and I. G. Macdonald. On the Adem relations. Topology, 21(3):329-332, 1982.
[Bre97] Glen E. Bredon. Topology and geometry, volume 139 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1997.
[Hat] Allen Hatcher. Spectral sequences in algebraic topology. Book project, available at http://www.math.cornell.edu/~hatcher/SSAT/SSATpage.html.
[MC01] John McCleary. A User's Guide to Spectral Sequences. 2nd edition, Cambridge University Press, 2001.
[Mil58] John Milnor. The Steenrod algebra and its dual. Ann. of Math. (2), 67:150-171, 1958.
[MT68] Robert E. Mosher and Martin C. Tangora. Cohomology operations and applications in homotopy theory. Harper & Row Publishers, New York, 1968.