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

Tuesday, 9:0011:00, room 009

Thursday, 9:0011:00, room 009

Friday, 11:0013:00, room 009
Office hours: Tuesday, 11:0013:00, office 302
Bibliography

A. Hatcher: Algebraic Topology

Edelsbrunner, Harer, Computational Topology

M. J. Greenberg, J. R. Harper, Algebraic Topology, A first course

R. Bott, L.W. Tu, Differential forms in algebraic topology

Lecture notes by E. Arbarello and R. Salvati Manni, Chapter 5

MacLane, Homology

J. Milnor, Morse Theory,

G. Carlsson, Topology and Data,

F. Chazal, B. Michele, An introduction to Topological Data Analysis: fundamental and practival aspects for data scientist, arxiv

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. MayerVietroris 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 2829
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 2829, 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 pathconnected space

4/10 Direct sum of complexes; homology and decomposition in pathconnected 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. MayerVietoris 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