Cohomology Operations and Applications (Graduate Studies A)

Organiser: Nora Ganter

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Course Description

Aimed at PhD and highly motivated Masters students, this class sets out to explore the notions of cohomology theories and their operations. The format of the class is freely inspired by the Kan-seminar, an intensive second year graduate course that has for decades formed the foundation of the graduate algebraic topology program at MIT. The lectures are given by teams of students, and I will attempt to team up the locals with those stuck overseas, so as to enable the community to grow together a little more. I will help preparing and polishing the presentations. It is an extremely valuable tradition that each talk is preceded by a practice talk, which is for you to schedule and attend as you like. The rigorously presented scientific part will be supplemeted by community and discussion sessions, which will be a mix of in person and online events. Delivery of the talks will be a combination of online and in person, hopefully often outdoors, with the outdoors presentations recorded for those who cannot make it.

Readings

R.E. Mosher and M.C. Tangora: Cohomology Operations and Applications in Homotopy Theory. Harper & Row, Publishers, New York-London 1968 x+214 pp.

J.F. Adams and M.F. Atiyah: K-theory and the Hopf invariant. Quart. J. Math. Oxford Ser. (2) 17 (1966), pp 31- 38.

Schedule

Talk	Topic	Reading
1	Introduction to Cohomology Operations	Mosher Tangora pp.1-7
2	Construction of the Steenrod Squares	Mosher and Tangora pp.12-18
3	Properties of the Squares	Mosher and Tangora pp.22-29
4	The Hopf Invariant	Mosher and Tangora pp.33-37
5	K-theory, Adams Operators and the Hopf Invariant Revisited	Adams and Atiyah
6	Vector Fields on Spheres	Mosher and Tangora pp.39-43
7	The Steenrod Algebra	Mosher and Tangora pp.45-56
8	Exact Couples and Spectral Sequences	Mosher and Tangora pp. 59-70
9	The Spectral Sequence of a Fibre Space	Mosher Tangora pp.73-80
10	Cohomology of the Eilenberg MacLane Spaces	Mosher and Tangora pp.83-91