Algebraic topology
Master 1, ENS de Lyon
January-April 2024

Announcements

  • Solutions of the final exam:
  • Solutions of the midterm exam:
  • The final exam is scheduled for Monday April 29th, from 2:00pm to 5:00pm, in Amphi A.
  • The midterm exam is scheduled for Tuesday March 5th, from 8:00am to 10:00am, in Amphi A.
  • The tutorial of February 13 is canceled. There will be an extra tutorial on February 9.
  • All the students attending the course should subscribe to it in the portail des études.

Schedule

Calendar (lectures, tutorials)

January 16: Introduction
Recap of general topology
Fundamental group
January 17: Brower's fixed point theorem in dimension 2
Theorem of the invariance of domain in dimension 2
Introduction to higher homotopy groups
Singular homology
January 23:
January 24:
January 30: Hurewicz theorem
Homology homomorphisms induced by continuous maps
Homotopy invariance of singular homology
Short and long exact sequences of chain complexes
January 31: Relative homology
Subdivision homomorphism
Theorem of small chains
Excision
February 6: Mayer-Vietoris long exact sequence
Homology of spheres
Local homology
Theorem of the invariance of domain
Brower's fixed point theorem
Topological degree
Hairy ball theorem
February 7:
February 9: (exceptional session)
February 13: (canceled session)
February 14:
February 20: CW complexes
CW decomposition of spheres and projective spaces
Cellular homology theorem
Homology of quotients and wedge sums
February 21:
March 12: Boundary homomorphism in cellular homology
Homology of real projective spaces
Euler characteristic
Singular homology with coefficients
Singular cohomology
March 13:
March 19: Properties of singular cohomology
Free resolutions
March 20:
Solution of the exercise on Poincaré's hypercube manifold:
March 26: Ext and Tor functors
Proof of the universal coefficient theorem
March 27:
Some solutions:
Avril 2: Cup product
Crossed cup product
Künneth theorem
Avril 3:
Some solutions:
April 9: Orientability of topological manifolds
Fundamental class
Cohomology with compact support
April 10:
April 16: Cap product
Poincaré duality
Proof of Poincaré duality:
April 17:
UMPA ENS de Lyon Université de Lyon