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

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

Section 1, page 7, exercises from1 to 5 ; Tuesday 8th October
Section 2 page 16, exercises 1 to 3 ; Tuesday 15th October
Section 3 page 28-29, from 1 to 14 except 7 ; Tuesday 29th October

Possible topics for the final presentation:

Spectral sequences and homology of fiber bundles. For instance, you can focus on Serre spectral sequence
J.P. Serre, Homologie Singulière des esapces fibres
Hatcher, Spectral sequences in algebraic topology
Group cohomology.
Kenneth S. Brown, Cohomology of groups, (The definition of group homology is in chapter 2)
J.P. Serre, Galois Cohomolgy (for the first read, you can try to read directly chapter 2, replacing profinite with finite)
S. Maclane, Homology, Chapter 4
A.I. Kostrikin, I. R. Shafarevich (Eds.), Algebra V, Homological Algebra, Chapter 2, especially 2.7
Morse theory
J. Milnor, Morse Theory
Topological Data Analysis
see for instance these two papers arxiv and arxiv

Lecture Schedule:

27/9 Introduction
1/10 Complexes of modules. Definition of the homology of topological spaces.
3/10 Categories and functuors. Homology of a point. Zero degree homology of a path-connected space
4/10 Direct sum of complexes; homology and decomposition in path-connected components. Five lemma
8/10 Homotopies of morphisms of complexes; homotopy invariance of the homology of topological spaces. Exercises.
10/10 Snake lemma
11/10 Relative homology of a pair and the associated long exact sequence
15/10 Relative homology and homology of the quotient: statement and applications. Homology of the spheres.
17/10 More about the homology of the sphere.Satement of the barycentric subdivision. Mayer-Vietoris exact sequence. Hopf fibration.
18/10 Homology of complex projective spaces. Exercises.
22/10 Excission. Proof of the theorem of the homology of the quotient. Exercises.
24/10 Symplicial complexes and symplicial homology.
25/10 NO CLASS
29/10 Naturality of the snake lemma. Isomorphism of symplicial and singular cohomology. Homology of the real projective plane
19/11 Topological Data Analysis. Exercises.
21/11 Exercises
22/11 Exerscies and preparation of final essays
26/11 Degree of of endomorphisms of a sphere. Exercises. CW complexes.
28/11 NO CLASS
29/11 NO CLASS
3/12 Tensor product of modules. Overview of Tor and Kunneth formula
5/12 4 hours Projective resolutions and preparation of final presentations.
6/12 Tor functors
10/12 Kunneth formula.
12/12 proof of barycentirc subdivsions
17/12 Serre spectral sequence
20/12 Group cohomology
7/1 Exercises. Wedge product and differential forms.
9/1 Morse theory
10/1 4 hours, room 311Exercise

Algebraic Topology (GE450)