## Math 237 Spectral Sequences - Spring 2020 at UCLA

Graduate topics course in algebraic topology (during a pandemic)

Syllabus

**Format**

Pre-recorded lectures with synchronous discussions during class periods.

Students gave final presentations in the last two weeks.

#### Video Lectures

Playlist of all videos

**Motivation and setup**

Video 1 - Intro and motivation

Video 2 - Graded rings and modules

Video 3 - Spectral sequence definition

Video 4 - 1st quadrant spectral sequence

Video 5 - Homology 1st quadrant spectral sequence

Video 6 - Unraveled exact couple (part 1)

Video 7 - Unraveled exact couple (part 2)

Video 8 - Unraveled exact couple (part 3)

Video 9 - Cohomology spectral sequence for a filtration

Video 10 - Unraveled exact couple (part 4)

Video 11 - Exact couple

**Bockstein spectral sequence**

Video 12 - Bockstein spectral sequence

Video 13 - Bockstein spectral sequence 1st application

Video 14 - Bockstein spectral sequence from a filtration

**Serre spectral sequence**

Video 15 - Serre spectral sequence

Video 16 - Serre spectral sequence sketch

Video 17 - Loop-path fibration

Video 18 - First computations (part 1)

Video 19 - First computations (part 2)

Video 20 - An application of the Serre spectral sequence

Video 21 - The Steenrod squares

Video 22 - The Steenrod algebra

Video 23 - Unstable modules over A

Video 24 - Cohomology of Eilenberg-MacLane spaces (part 1)

Video 25 - Cohomology of Eilenberg-MacLane spaces (part 2)

Video 26 - Cohomology of Eilenberg-MacLane spaces (part 3)

Video 27 - Cohomology of Eilenberg-MacLane spaces (part 4)

Video 28 - Cohomology of Eilenberg-MacLane spaces (part 5)

**Stable homotopy groups**

Video 29 - Calculating first stable stem

Video 47 - Stable stems are finite

Video 48 - Computing more stable stems (part 1)

Video 49 - Computing more stable stems (part 2)

**Double complexes**

Video 30 - Double complex

Video 31 - Double complex spectral sequence

Video 32 - Some applications (part 1)

Video 33 - Some applications (part 2)

Video 34 - Grothendieck spectral sequence

Video 35 - Base change

Video 36 - Construction of Grothendieck spectral sequence (part 1)

Video 37 - Construction of Grothendieck spectral sequence (part 2)

**Convergence**

Video 38 - Cohomology spectral sequence for a filtration (part 1)

Video 39 - Cohomology spectral sequence for a filtration (part 2)

Video 40 - Convergence (part 1)

Video 41 - Convergence (part 2)

Video 42 - Convergence (part 3)

Video 43 - Convergence (part 4)

Video 44 - Convergence (part 5)

Video 45 - Convergence (part 6)

Video 46 - Examples

#### Resources

Extra chapter on spectral sequences - Hatcher

*Spectral sequences: friend or foe?* - Vakil

*Lecture notes in algebraic topology* - Davis and Kirk

*Cohomology operations and applications in homotopy theory* - Mosher and Tangora

*A user's guide to spectral sequences* - McCleary

*A user's guide to the Bockstein spectral sequence* - Palmieri

*Spectral sequences and an application* - Orlich

*Conditionally convergent spectral sequences* - Boardman

*An introduction to homological algebra* - Weibel

#### Suggested Presentation Topics

Bockstein spectral sequence and H-spaces

Proof of cohomological Serre spectral sequence with algebra structure

Higher structure in Serre spectral sequence (e.g. coalgebras and/or Hopf algebras)

Proof of existence of Steenrod squares and properties

Computation of K(Z/2,n) with integral coefficients using Bockstein spectral sequence

Odd prime Steenrod operations

Proof of Thom isomorphism using Serre spectral sequence

Computations in and/or construction of:

The Serre spectral sequence with local coefficients

The Bökstedt spectral sequence

The Atiyah-Hirzebruch spectral sequence

The Künneth spectral sequence

The change-of-rings spectral sequence

The Cartan-Leray spectral sequence

The slice spectral sequence

The Bousfield-Kan spectral sequence

The homotopy fixed-point spectral sequence