Cohomology Operations
Spring 2024
Cohomology Operations
Spring 2024
A motivational phrase (otherwise known as a life motto) for algebraic topology is
Topology is hard, algebra is easy.
While algebra is by no means easy, the tools of algebraic topology can often translate a seemingly intractable topological problem into a much simpler algebraic one. Better yet, the more sophisticated the tool, the more breadth of questions we can attack via this method. The fundamental group is the first invariant one encounters, and is very powerful. Moving in a more algebraic direction are homology groups, and one step further is the cohomology ring. With each step we get more structure to leverage in our classification goals.
As it turns out, cohomology is host to much more structure than just the cup product. There are a plethora of (stable) cohomology operations. Working with coefficients in F2 from here on (and forever), the set of all these cohomology operations forms an algebra called the Steenrod algebra, and the cohomology of any space is naturally a module over this algebra. It is another algebraic tool at our disposal, and its use far supersedes any previous tool.
This seminar will investigate cohomology operations and the Steenrod algebra. We will construct the Steenrod algebra and explore a plethora of its applications in algebraic topology. The tools involved are a mix of homological algebra, point set and algebraic topology, and combinatorics. The seminar will primarily follow the first few section of the wonderful book Cohomology Operations by Mosher and Tangora.
March 29, 2024: Introduction to cohomology operations
Jackson Morris
April 4, 2024: Construction of the Steenrod squares
Jay Reiter
April 12, 2024: Properties of the squares
Michael Zeng
April 19, 2024: The Hopf invariant
Jay Reiter
April 26, 2024: The Steenrod algebra and its dual
Alex Waugh
May 3, 2024: Introduction to spectral sequences
Jackson Morris
May 9, 2024: Worksheet
May 24, 2024: Worksheet