Algebraic topology
Master 1, ENS de Lyon
January-April 2022

Schedule

Exams

  • Midterm : March 10, 2022, 8:00am - 10:00am.
  • Final : April 25, 2022, 1:30pm - 3:30pm.

Calendar (lectures, tutorials)

January 11: Exercises:
January 13: Exercises:
January 18: Introduction
Recap of general topology
Fundamental group
Brower's fixed point theorem in dimension 2
Theorem of the invariance of domain in dimension 2
Introduction to higher homotopy groups
January 20: Singular homology (ordinary and reduced)
Hurewicz map
January 25: Hurewicz theorem in degree 1
Homology homomorphisms induced by continuous maps
Homotopy invariance of singular homology
Short and long exact sequences of chain complexes
January 27: This tutorial is canceled due to a strike
Exercises:
February 1: Relative homology
Subdivision homomorphism
February 3: Exercises:
February 8: Theorem of small chains
Excision
Mayer-Vietoris long exact sequence
Homology of spheres
Theorem of the invariance of domain
Brower's fixed point theorem
February 10: Exercises:
February 15: Topological degree
Hairy ball theorem
CW complexes
February 17: Exercises:
March 1: Cellular homology theorem
Euler characteristic
Singular homology with coefficients
March 3: Exercises:
March 15: Singular cohomology
Free resolutions
Universal coefficient theorem
March 17: Exercises:
March 22: Proof of the universal coefficient theorem
Cup product
March 24: Exercises:
March 29: Crossed cup product
Künneth theorem
Topological manifolds
March 31: Exercises:
April 5: Orientability of topological manifolds
Fundamental class
Compactly supported cohomology
April 7: Exercises:
April 12: Cap product
Poincaré duality
April 14: Exercises:
UMPA ENS de Lyon Université de Lyon