In Spring 2021, Lennart Meier and I will be running a seminar for MSc students on spectral sequences. The goal is to get acquainted with spectral sequences as a computational tool in algebraic topology. The main examples we will consider are the Serre spectral sequence and the Adams spectral sequence. Along the way we will discuss some of the basics of stable homotopy theory.
Here is the syllabus. The seminar will take place on Fridays, 3-5pm, through Zoom. Starting April 30, the seminar will move to the timeslot 1-3pm, still on Fridays.
Feb 12, Gijs: Introduction and overview.
Feb 19, Sven: Review of basic homotopy theory. Fibrations, Eilenberg-MacLane spaces, Postnikov towers, loop spaces. Sven's notes and exercises.
Feb 26, Gerben: Spectral sequences, definitions and first examples. Exact couples. Gerben's exercises.
Mar 5, Antonie: The spectral sequences of a filtered complex and of a double complex. The Dress construction of the Serre spectral sequence. Antonie's notes and exercises.
Mar 12, Max: More on the Serre spectral sequence. Local coefficients, multiplicativity. Max' exercises.
Mar 19, Martijn: First applications of the Serre spectral sequence. The Hurewicz theorem, some homotopy groups of spheres. Martijn's exercises.
Mar 26, Wessel: Spectral sequences in homological algebra. Tor and Ext, the Künneth and Universal Coefficient spectral sequences. Wessel's notes and exercises.
Apr 2: No seminar (Easter).
Apr 9, Stan: Serre class theory. The mod C Hurewicz theorem, homotopy groups of spheres are finitely generated. Stan's notes and exercises.
Apr 16, Max: The mod C Whitehead theorem, rational homotopy groups of spheres, the first p-torsion in the homotopy groups of spheres. Max' exercises.
Apr 23, Sven: Cohomology of Eilenberg-MacLane spaces. Sven's notes and exercises.
Apr 30, Lennart: Introduction to stable homotopy theory, I. Lennart's notes.
May 7, Lennart: Introduction to stable homotopy theory, II. Lennart's notes.
May 21, Antonie: Setting up the Adams spectral sequence. Antonie's exercises and notes.
May 28, Stan: The E_2-page of the Adams spectral sequence and the first few stable stems. Stan's notes and exercises.
June 4, Gerben: Multiplicative structure in the Adams spectral sequence. Gerben's notes and exercises.
June 11, Martijn: Adams filtration, Hopf invariant one, and the Adams spectral sequence for ko.