【HSE】An Introduction to Cobordism Theory 配边理论

发布人

https://www.hse.ru/en/edu/courses/476650478
Alexei Gorinov
/playlist?list=PLq3E5oubNNoD888S6OstxN0whZbWAcvdi

DESCRIPTION: 
The starting point of cobordism theory is the question whether or not one smooth manifold is the boundary of another. This question and a few similar ones can be answered using homotopy theory. Vice versa, some of the strongest known results of homotopy theory make essential use of cobordisms of some type or another. 
We will begin by looking at the Pontrjagin – Thom construction, which allows one to reduce the above question and its variants (oriented, non-oriented, framed etc.) to calculating the homotopy groups of the corresponding Thom spectrum. Then we will study the classical applications. In particular, we will see that a smooth manifold bounds another smooth manifold if and only if all its Stiefel – Whitney numbers vanish. After that we will focus on complex cobordism and applications to homotopy theory. 

PREREQUISITES: 
Smooth manifolds as covered in the compulsory course; homology and cohomology as covered in Algebraic topology 1 or the first three chapters of Hatcher’s Alegbraic topology. 

SYLLABUS: 
1. Examples of bordisms: oriented, non-oriented, complex and framed bordisms. 
2. The Pontrjagin – Thom theorem. 
3. Spectra and their homotopy groups: a reminder. The Thom spectra. 
4. The Adams spectral sequence. 
5. Applications of the Adams spectral sequence to the calculation of bordism groups. 
6. The Hurewicz homomorphism. 
7. Orientations of vector bundles with respect to multiplicative cohomology theories. Complex oriented theories. 
8. Formal group laws and Quillen’s theorem. 
9. Cohomology operations and the Landweber-Novikov theorem. 
10. (*) Brown – Peterson spectra. 
11. (*) Landweber’s exact functor theorem. 
12. (*) Elliptic cohomology. Topological modular forms. 
13. (*) Chromatic spectral sequence and Morava’s 𝐾-theories. 

TEXTBOOKS: Haynes Miller, Vector fields on spheres etc. (online notes). 
仅供学习交流使用，侵删

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【HSE】An Introduction to Cobordism Theory 配边理论

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