Algebraic Topology (GE450)

Accademic year: 2019/2020 - First Semester
Lecturer: Giulio Codogni

On Wednesday 15th of January, seminar on Topological Data Analysis, from 14:30 to 16:30, room 009

On Friday 10th of January there will be 4 hours lectures, from 9:00 to 11:00 room 311, from 11:00 to 13:00 room  009

Schedule

Office hours: Tuesday, 11:00-13:00, office 302

Bibliography

  1. A. Hatcher: Algebraic Topology
  2. Edelsbrunner, Harer,  Computational Topology
  3. M. J. Greenberg, J. R. Harper, Algebraic Topology, A first course
  4. R. Bott, L.W. Tu, Differential forms in algebraic topology
  5. Lecture notes by E. Arbarello and R. Salvati Manni, Chapter 5
  6. MacLane, Homology
  7. J. Milnor, Morse Theory,
  8. G. Carlsson, Topology and Data,
  9. F. Chazal, B. Michele, An introduction to Topological Data Analysis: fundamental and practival aspects for data scientist, arxiv
  10. Fast and Accurate Tumor Segmentation of Histology Images using Persistent Homology and Deep Convolutional Features, arxiv

This course is partially inspired by the course "Topics in Geometry - Computational Topology", by G. Székelyhidi

Final Course Syllabus:

The course was focused on homological algebra and homology of topological spaces. We manly followed the book “Algebraic Topology” by Hatcher, covering in details Chapters 2.1, 2.2 , 3.A and 3.B. 

 

Homological algebra. complexes of modules, morphisms, homotopies, and homology. Fives lemma. Snake lemma. Examples of short and long exacts sequences. Directs sums and tensor products of modules and complexes of modules. Tor functors, using projective resolutions. Kunneth formula for complexes over a PID. 

 

Homology of topological spaces. Singular homology, definitions and examples. Mayer-Vietroris exact sequence. Homology of pairs, Excision Theorem. Universal Coefficient Theorem. Kunneth formula. Simplicial and cellular homology, comparison with singular homology. Examples: spheres, real and complex projective spaces.

 

Exercises: We corrected in class the following exercises from taken from the Lecture notes by E. Arbarello and R. Salvati Manni, Chapter 5: 1 to 5 section 1 page 7, 1 to 3 section 3 page 16, and 1 to 14 except 7 section 5 pages 28-29

 

Topological Data Analysis: Definition of Chech complex associated to a finite metric space. Persistent homology. Barcodes. ( following part of F. Chazal, B. Michele, An introduction to Topological Data Analysis: fundamental and practival aspects for data scientist, arxiv , Sections 2 and 5)

 

In the last part of the course we briefly surveyed the following topics: cohomology and Poincaré duality, following Chapter 3 of Hatcher; manifolds, differential forms and de Rham cohomology, cohomology of sheaves. Serre spectral sequence for fiber bundles, group cohomology, Morse theory.

Language: English or Italian, to be decided together with the students.

Exam: Homeworks, presentation of at least one execrcise at the blackboard, and final presentaton in the form of a seminar.

Homeworks Exercises are taken from the Lecture notes by E. Arbarello and R. Salvati Manni, Chapter 5

Possible topics for the final presentation:

Lecture Schedule:

  1. 27/9 Introduction
  2. 1/10 Complexes of modules. Definition of the homology of topological spaces.
  3. 3/10 Categories and functuors. Homology of a point. Zero degree homology of a path-connected space
  4. 4/10 Direct sum of complexes; homology and decomposition in path-connected components. Five lemma
  5. 8/10 Homotopies of morphisms of complexes; homotopy invariance of the homology of topological spaces. Exercises.
  6. 10/10 Snake lemma
  7. 11/10 Relative homology of a pair and the associated long exact sequence
  8. 15/10 Relative homology and homology of the quotient: statement and applications. Homology of the spheres.
  9. 17/10 More about the homology of the sphere.Satement of the barycentric subdivision. Mayer-Vietoris exact sequence. Hopf fibration.
  10. 18/10 Homology of complex projective spaces. Exercises.
  11. 22/10 Excission. Proof of the theorem of the homology of the quotient. Exercises.
  12. 24/10 Symplicial complexes and symplicial homology.
  13. 25/10  NO CLASS
  14. 29/10 Naturality of the snake lemma. Isomorphism of symplicial and singular cohomology. Homology of the real projective plane
  15. 19/11 Topological Data Analysis. Exercises.
  16. 21/11 Exercises
  17. 22/11 Exerscies and preparation of final essays 
  18. 26/11 Degree of of endomorphisms of a sphere. Exercises. CW complexes.
  19. 28/11 NO CLASS
  20. 29/11 NO CLASS
  21. 3/12 Tensor product of modules. Overview of Tor and Kunneth formula
  22. 5/12 4 hours Projective resolutions and preparation of final presentations.
  23. 6/12 Tor functors
  24. 10/12 Kunneth formula.
  25. 12/12 proof of barycentirc subdivsions
  26. 17/12 Serre spectral sequence
  27. 20/12 Group cohomology
  28. 7/1 Exercises. Wedge product and differential forms.
  29. 9/1 Morse theory
  30. 10/1 4 hours