Class Notes
| Key Topics | Notes | Podcast |
|---|---|---|
| Review of Homological Algebra | ||
| Homological Algebra II: Derived Functors | ||
| Easy Introduction to Spectral Sequences | ||
| Overview of Serre Spectral Sequence | ||
| Gysin Sequence and Hochschild-Serre Spectral Sequence | MP3 | |
| Steenrod Operations: Axioms and Properties | MP3 | |
| Construction of the Squares | MP3 | |
| Cohomology of Eilenberg-MacLane Spaces at 2 | MP3 | |
| Massey Products and the Steenrod Algebra | MP3 | |
| Postnikov Towers | MP3 | |
| Nitty-Gritty of Spectral Sequences: Exact Couples | MP3 | |
| Bockstein Spectral Sequences | ||
| Adams Filtration and the Adams Spectral Sequence | ||
| Warm-Up for the Adams Spectral Sequence: ko-Homology | MP3 | |
| Computing Ext Groups: May Spectral Sequence | ||
| Secondary Operations | ||
| More Ext & Adams Differentials | MP3 | |
| Hopf Invariant 1 Problem | MP3 | |
| "User Generated Computations" | ||
| 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 | ||
| Slice Spectral Sequence I | ||
| Slice Spectral Sequence II |
Homework
| Description | File | Sols |
|---|---|---|
| Ext and Tor Problems | ||
| Practice with the Serre Spectral Sequence | ||
| Steenrod Squares | ||
| Postnikov Towers & Homotopy | ||
| Adams Spectral Sequence I | ||
| Adams Spectral Sequence II |
Foundational Papers
| Author & Title | File |
|---|---|
Serre - Homologie Singuliere des Espaces FibresThis paper is Serre's Annals exposition of his eponymous spectral sequence and the basic properties. |
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Steenrod, Thomas - Cohomology Operations derived from Cyclic GroupsThe paper provides an analysis of the Steenrod squares and reduced power operations. |
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Milnor - The Steenrod Algebra and its DualThis beautiful paper describes the Hopf algebra structure of the Steenrod algebra and the dual Steenrod algebra. |
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Serre - Cohomology modulo 2 des complexes d'Eilenberg-MacLaneThis is Serre's computation of the cohomology of Eilenberg-MacLane spaces. |
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Serre - Groupes d'homotopie et classes de groupes abeliensThis is an introduction to Serre classes and computations of homotopy groups using them. |
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Adams - On the Structure and Applications of the Steenrod AlgebraAdams' construction of the Adams spectral sequence and an exposition of some of its basic properties. |
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May - The Cohomology of the Steenrod AlgebraThis is an unpublished book by May describing the May spectral sequence and computations therewith. |
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Adams - On the Non-Existence of Elements of Hopf Invariant OneThis 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! |
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Liulevicius - The Factorization of Cyclic Reduced Powers by Secondary Cohomology OperationsThis is the odd primary analogue of Adams' Hopf Invariant One paper. |
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Bruner, McClure, May, Steinberg - $H_\infty$-Ring Spectra and their ApplicationsThis book is a detailed and complete analysis of spectra with power operations and their relations with such things as the Adams spectral sequence. |
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Eilenberg, Moore - Homology and Fibrations 1This paper establishes the basics of the Eilenberg-Moore spectral sequence and the functor Cotor. |
Pictures
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The Serre Spectral Sequence for Loops S^3. |
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The Serre Spectral Sequence for U(3). Circled classes are algebra generators. |
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The Serre Spectral Sequence for 2-frames in R5. The drawn differential is multiplication by 2. |
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The Serre Spectral Sequence for 2-frames in R6. |
| These are all made using Tilman Bauer's SSEQ package for LaTeX. |