Master II Lecture

Homotopy Theories

(November-December 2021)


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, 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 Emmanuel Wagner "Théorie de l'homologie" (September-October 2021); it will open the doors to the one of Najib Idrissi Homotopie II" (January-February, 2022) and to the one of Muriel Livernet "Catégories supérieures" (March-April 2022).

Lecture Notes

The notes of the course will be available here:    (Final version, 1/1/21)

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

Layout
  1. Homotopy theory of topological spaces
  2. Simplicial homotopy theory

Worksheet

  • Worksheet 1     [Version 2021]
  • Worksheet 2     [Version 2021]
  • Worksheet 3     [Version 2021]
  • Worksheet 4     [Version 2021]
  • Worksheet 5     [Version 2021]
  • Worksheet 6     [Version 2021]

Exam (2020)

  • English version     
  • Version française    
  • Corrigé    

Exam (2018)

  • English version     
  • Version française    
  • Corrigé    

Analysis Situs

  • Analysis situs, superbe site dévolu aux travaux de Poincaré sur la topologie algébrique des variétés.

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 Emmanuel Wagner's course: category, functor, adjunction, (co)limits, topological space, homeomorphism.

Professor

       Bruno Vallette (lectures/exercise sessions)



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Last updates : September 6th, 2021.