Spectral Sequences in Algebraic Topology
The computation of the homotopy groups of spheres, especially the stable stems, guides this course through the Serre, Adams, and Eilenberg–Moore spectral sequences. Starting with filtrations and fibrations, we develop transgression, cohomology operations, Postnikov towers, and torsion calculations. We then pass to spectra, the Steenrod algebra, Adams resolutions, and explicit stable homotopy computations, culminating in the Hopf invariant one problem. The Eilenberg–Moore sequence connects homotopy pullbacks with bar constructions and leads to calculations of iterated loop spaces and an obstruction to realizing a Steenrod module as the cohomology of a space. Following Ravenel’s notes and incorporating Hill’s examples and applications, the course emphasizes complete calculations, geometric meaning, and the passage from spectral-sequence pages to their targets.
讲师
日期
2026年09月23日 至 12月23日
位置
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| 周三,周五 | 13:30 - 15:05 | Shuimo-LG28 | - | - | - |
课程大纲
Lecture 1: The sphere problem and the Serre machine
Lecture 2: Products and exact sequences
Lecture 3: Groups and homogeneous spaces
Lecture 4: Loop spaces and Borel’s theorem
Lecture 5: Construction of cohomology operations
Lecture 6: The Steenrod algebra and K(A, n)
Lecture 7: Massey products and odd primes
Lecture 8: Postnikov towers and Serre’s program
Lecture 9: Degree maps and Bocksteins
Lecture 10: The spectral-sequence formalism
Lecture 11: Spectra and stable maps
Lecture 12: Adams resolutions
Lecture 13: The Adams spectral sequence
Lecture 14: Ext and stable mapping problems
Lecture 15: Milnor’s theorem and cobar
Lecture 16: May, change of rings, ku and ko
Lecture 17: Ext charts and secondary information
Lecture 18: Stable stems and extensions
Lecture 19: Coefficient and Moore spectra
Lecture 20: Hopf invariant one
Lecture 21: The Eilenberg–Moore sequence
Lecture 22: Computations with EMSS
Lecture 23: Iterated loop spaces
Lecture 24: Operations and realization
Lecture 2: Products and exact sequences
Lecture 3: Groups and homogeneous spaces
Lecture 4: Loop spaces and Borel’s theorem
Lecture 5: Construction of cohomology operations
Lecture 6: The Steenrod algebra and K(A, n)
Lecture 7: Massey products and odd primes
Lecture 8: Postnikov towers and Serre’s program
Lecture 9: Degree maps and Bocksteins
Lecture 10: The spectral-sequence formalism
Lecture 11: Spectra and stable maps
Lecture 12: Adams resolutions
Lecture 13: The Adams spectral sequence
Lecture 14: Ext and stable mapping problems
Lecture 15: Milnor’s theorem and cobar
Lecture 16: May, change of rings, ku and ko
Lecture 17: Ext charts and secondary information
Lecture 18: Stable stems and extensions
Lecture 19: Coefficient and Moore spectra
Lecture 20: Hopf invariant one
Lecture 21: The Eilenberg–Moore sequence
Lecture 22: Computations with EMSS
Lecture 23: Iterated loop spaces
Lecture 24: Operations and realization
参考资料
1. https://www.sas.rochester.edu/mth/sites/doug-ravenel/oldcourses/549s18/notes/index.html
2. https://www.math.ucla.edu/~mikehill/Teaching/Math885.html
3. Douglas Ravenel: Complex cobordism and stable homotopy groups of spheres
4. Robert Mosher and Martin Tangora: Cohomology operations and applications in homotopy theory
5. John McCleary: A user's guide to spectral sequences
2. https://www.math.ucla.edu/~mikehill/Teaching/Math885.html
3. Douglas Ravenel: Complex cobordism and stable homotopy groups of spheres
4. Robert Mosher and Martin Tangora: Cohomology operations and applications in homotopy theory
5. John McCleary: A user's guide to spectral sequences
视频公开
公开
笔记公开
公开
语言
英文
讲师介绍
本科和博士毕业于中国科学技术大学，之前是清华大学丘成桐数学科学中心博士后，现在是北京雁栖湖应用数学研究院助理研究员，研究兴趣是用拓扑方法（cobordism）研究理论物理（anomaly）。