Master II Lecture

Homotopy Theories

(November-December 2024)

Abstract

The goal of this lecture will be to present various "concrete" homotopy theories. We will start with the classical homotopy theory of topological spaces (higher homotopy groups, cofibration and fibrations, cellular complexes, Whitehead and Hurewicz theorems). Then we will move to the homotopy theory of simplicial sets (definitions, simplex category, adjunction and cosimplicial objects, examples, fibrations, Kan complexes, and simplicial homotopy).

This course will directly follow the one of Bernhard Keller "Homologie, cohomologie et faisceaux" (September-October 2024) and it will be pursued further by the course of Grégory Ginot "Homotopie II" (January-February, 2024). It is needed to understand the specialised courses of Benoit Fresse "Operads, graph complexes and applications" (March-April, 2024) and Maria Yakerson " How homotopy theory helps algebraic geometry" (March-April, 2024).

Organisation

The courses will take place

Monday 2.30pm-5.30pm (Sophie Germain 2011 40): November 4, 18, 25, December 2, 9.
Wednesday and Friday 9am-12am (Jussieu 16/26 1.13): December 6, 11.
Thursday 9am-12am (Jussieu 15/25 1.01): November 7, 14, 21, 28.
Friday 4pm-7pm (Sophie Germain 1021): December 13.

Lecture Notes

The notes of the course are available here:

(version of December 11, 2024)

Layout

Homotopy theory of topological spaces
Simplicial homotopy theory

Worksheet

Worksheet 1
Worksheet 2
Worksheet 3
Worksheet 4
Worksheet 5
Worksheet 6

YouTube links (2020 course)

Course 1 [4 november]: https://youtu.be/tFEjsTGzVfk (1/2) & https://youtu.be/NV_Q8StHP6Y (2/2)
Course 2 [5 november]: https://youtu.be/W_Ze-h04fHo (1/2) & https://youtu.be/zChw5rzs1IY (2/2)
Course 3 [11 november]: https://youtu.be/O3rx2rHJZPM (1/2) & https://youtu.be/jR8yXHouaFk (2/2)

Course 4 [12 november]: https://youtu.be/7s8WEuuPrw0 (1/2) & https://youtu.be/KG3VEW5P09w (2/2)
Course 5 [18 november]: https://youtu.be/zrfybhj5xqk
Course 6 [19 november]: https://youtu.be/O6mTvYjDczY (1/2) & https://youtu.be/YWSsAR30IaY (2/2)

Course 7 [25 november]: https://youtu.be/5jTArhCNn4E

Course 8 [26 november]: https://youtu.be/tAdSwbQbqI8 (1/2) & https://youtu.be/ZMvw8b0SY78 (2/2)

Course 9 [2 december]: https://youtu.be/Z6l6vp-r8tk

Course 10 [3 december]: https://youtu.be/JjcYRKKT85Y

Course 11 [9 décember]: https://youtu.be/zFleVAuHq3E

Course 12 [10 december]: https://youtu.be/xz-nJ7NF59o (1/2) & https://youtu.be/T57GHF0Yy2M (2/2)

Course 13 [16 december]: https://youtu.be/66mFKD_vfa8

Exam (2024)

English version
Version française
Corrigé

Exam (2021)

English version
Version française
Corrigé

Exam (2020)

English version
Version française
Corrigé

Exam (2018)

English version
Version française
Corrigé

Beyond the lectures

Analysis situs: great website dedicated to the works of Poincaré on the algebraic topology of manifolds.
Voyages aux pays des maths: episode of this great series (watch the other ones!) dedicated to the Poincaré conjecture.

References

Algebraic Topology, Tammo tom Dieck, EMS Textbooks in Mathematics, 2008.
A concise course in Algebraic Topology, J. Peter May, Chicago Lectures in Mathematics, 1999.
Algebraic Topology, Allen Hatcher, Cambridge University Press, 2001.

An elementary illustrated introduction to simplicial sets, Greg Friedman, arXiv:0809.4221, 2008.
Simplicial homotopy theory, Paul G. Goerss and John F. Jardine, Progress in Mathematics, 2009.
Simplicial homotopy theory, Edward B. Curtis, Advances in Mathematics 6, 107-209, 1971.
Simplicial Objects in Algebraic Topology, J. Peter May, Chicago Lectures in Mathematics, 1992.

Prerequisistes

From Bernhard Keller's course: category, functor, adjunction, (co)limits, topological space, homeomorphism.

Follow-up

Homotopie II by Grégory Ginot (Master 2 Mathematics, Fundamental Course II, January-February 2024).
Operads, graph complexes and applications by Benoit Fresse (Master 2 Mathematics, Specialized Course, March-April 2024).
How homotopy theory helps algebraic geometry by Maria Yakerson (Master 2 Mathematics, Specialized Course, March-April 2024).

Seminars

Seminar of the Algebraic Topology team (Université Sorbonne Paris Nord).

Professor

Bruno Vallette (lectures/exercise sessions)

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Last updates : January 9, 2025.