T,Th 11-12:15

Kerchof 326

By Appointment

# Class Notes

Key Topics Notes Podcast
Review of Homological Algebra PDF
Homological Algebra II: Derived Functors PDF
Easy Introduction to Spectral Sequences PDF
Overview of Serre Spectral Sequence PDF
Gysin Sequence and Hochschild-Serre Spectral Sequence PDF MP3
Steenrod Operations: Axioms and Properties PDF MP3
Construction of the Squares PDF MP3
Cohomology of Eilenberg-MacLane Spaces at 2 PDF MP3
Massey Products and the Steenrod Algebra PDF MP3
Postnikov Towers PDF MP3
Nitty-Gritty of Spectral Sequences: Exact Couples MP3
Bockstein Spectral Sequences PDF
Warm-Up for the Adams Spectral Sequence: ko-Homology PDF MP3
Computing Ext Groups: May Spectral Sequence
Secondary Operations
More Ext & Adams Differentials PDF MP3
Hopf Invariant 1 Problem PDF MP3
"User Generated Computations" PDF
Eilenberg-Moore Spectral Sequences MP3
Computations with the EMSS MP3
Geometry and the EMSS MP3
Computations Part II: Differentials and Products
Guest Lecture: Nick Kuhn PDF
Slice Spectral Sequence I
Slice Spectral Sequence II

# Homework

Description File
Ext and Tor Problems PDF
Practice with the Serre Spectral Sequence PDF
Steenrod Squares PDF
Postnikov Towers & Homotopy PDF

# Foundational Papers

Author & Title File

## Serre - Homologie Singuliere des Espaces Fibres

This paper is Serre's Annals exposition of his eponymous spectral sequence and the basic properties.

PDF

## Steenrod, Thomas - Cohomology Operations derived from Cyclic Groups

The paper provides an analysis of the Steenrod squares and reduced power operations.

PDF

## Milnor - The Steenrod Algebra and its Dual

This beautiful paper describes the Hopf algebra structure of the Steenrod algebra and the dual Steenrod algebra.

PDF

## Serre - Cohomology modulo 2 des complexes d'Eilenberg-MacLane

This is Serre's computation of the cohomology of Eilenberg-MacLane spaces.

PDF

## Serre - Groupes d'homotopie et classes de groupes abeliens

This is an introduction to Serre classes and computations of homotopy groups using them.

PDF

## Adams - On the Structure and Applications of the Steenrod Algebra

Adams' construction of the Adams spectral sequence and an exposition of some of its basic properties.

PDF

## May - The Cohomology of the Steenrod Algebra

This is an unpublished book by May describing the May spectral sequence and computations therewith.

PDF

## Adams - On the Non-Existence of Elements of Hopf Invariant One

This is Adams' original proof of the non-existence of Hopf invariant one elements beyond dimension 7. He explicitly decomposes the primary operations into secondary ones!

PDF

## Liulevicius - The Factorization of Cyclic Reduced Powers by Secondary Cohomology Operations

This is the odd primary analogue of Adams' Hopf Invariant One paper.

PDF

## Bruner, McClure, May, Steinberg - $H_\infty$-Ring Spectra and their Applications

This book is a detailed and complete analysis of spectra with power operations and their relations with such things as the Adams spectral sequence.

PDF

## Eilenberg, Moore - Homology and Fibrations 1

This paper establishes the basics of the Eilenberg-Moore spectral sequence and the functor Cotor.

PDF

# Pictures

 The Serre Spectral Sequence for Loops S^3. The Serre Spectral Sequence for U(3). Circled classes are algebra generators. The Serre Spectral Sequence for 2-frames in R5. The drawn differential is multiplication by 2. The Serre Spectral Sequence for 2-frames in R6. These are all made using Tilman Bauer's SSEQ package for LaTeX.