Algebraic topology
Master 1, ENS de Lyon
January-April 2023

Announcements

  • The final exam will take place on April 25, from 9:00am to 12:00am, in Amphi A.
  • The midterm exam will take place on February 28, from 8:00am to 10:00am, in Amphi A.
  • On Monday, February 6, we will have a tutorial at the usual scheduled time 8:00am, and we will also have an extra lecture from 5:00pm to 7:00pm in room 435.
  • On Monday, January 30, we will have a tutorial at the usual scheduled time 8:00am, and we will also have an extra lecture from 5:00pm to 7:00pm in room 435.
  • The lecture of January 16 and the tutorial of Tuesday January 17 are canceled.
  • All the students attending the course should subscribe to it in the portail des études.

Schedule

Calendar (lectures, tutorials)

January 9: 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 10: Exercises:
January 16: Canceled
January 17: Canceled
January 23: Exercises:
Solutions:
January 24: Singular homology
Hurewicz theorem
January 30: Exercises:
January 30: (extra lecture, from 5:00pm to 7:00pm, room 435)
Homology homomorphisms induced by continuous maps
Homotopy invariance of singular homology
Short and long exact sequences of chain complexes
Relative homology
January 31: Subdivision homomorphism
Theorem of small chains
Excision
Mayer-Vietoris long exact sequence
February 6: Exercises:
February 6: (extra lecture, from 5:00pm to 7:00pm, room 435)
Homology of spheres
Local homology
Theorem of the invariance of domain
Brower's fixed point theorem
Topological degree
Hairy ball theorem
CW complexes
February 7: CW decomposition of spheres and projective spaces
Cellular homology theorem
February 20: Exercises:
February 21: Homology of quotients and wedge sums
Boundary homomorphism in cellular homology
Homology of real projective spaces
Euler characteristic
March 6: Exercises:
March 7: Singular homology with coefficients
Singular cohomology
Free resolutions
March 13: Exercises:
March 14: Proof of the universal coefficient theorem
March 20: Exercises:
March 21: Cup product
Crossed cup product
Künneth theorem
March 27: Exercises:
March 28: Orientability of topological manifolds
Fundamental class
April 3: Exercises:
April 4: Cohomology with compact support
Cap product
Poincaré duality
April 17: Exercises:
April 18: Exercises:
UMPA ENS de Lyon Université de Lyon