Algebraic Topology, Master Degree, 6
Lectures: prof. Gilberto Bini
- Tuesday 10:30 am - 12:30 am, Aula 5
- Thursdau 3:30 pm - 5:30 am, Aula 9
- Office Hours: by appointment, 2045 sottotetto
This course is designed to present some basic
tools of Algebraic
Topology, such as singular homology and cohomology and their
relationship. For these purposes, we will recall some fundamental
topics as CW-complexes, homotopy theory and covering spaces.
Time permitting, we will introduce persistent homology and go over some
of its recent applications to various areas.
Please find below a list of problems taken from the textbook [S]. You
should choose 10 of them and return the homework at most one week
before you take the oral exam. In addition to some questions about the
chosen problems, the exam will go over the material covered in the
Exercise 1.1.19: (iv), (vii), (v),
- Exercise 1.2.33: (v), (xii), (xiii);
- Exercise 1.3.11: (iv);
- Exercise 1.5.19: (ii), (vii);
- Exercise 1.9: 3, 5, 7, 21, 28;
- Exercise 3.1.7: (iv);
- Exercise 3.5.23: (ii);
- Exercise 4.2.27: (ii), (iii), (iv), (v);
- Exercise 5.5: 5,6;
- Exercise 6.2.20: (i), (viii);
- Exercise 7.4.13: (iii);
- Exercise 8.1.21: (i);
- Exercise 8.2.19: (i), (v);
[GH] M. J. Greenberg, J. R.
Harper, Algebraic Topology.
A First Course, The Benjamin/Cummings Publishing Company, 1981.
[EH] H. Edelsbrunner, J. Harer, Computational topology.
An Introduction, AMS, Providence, 2009
[H] A. Hatcher, Algebraic Topology, online version.
[S] A. R. Shastri, Basic Algebraic Topology, CRC Press,
24 September 2015: A short summary
of the course. A
review of homotopy and first fundamental group. The homology of
complexes. CW- complexes.
- 29 September 2015: Complex and real projective space as CW-complexes. The Euler characteristic of a CW-complex.
- 1 October 2015: Some elements of homological algebra (the Snake lemma, the five lemma, etc.)
- 6 October 2015: Axioms for reduced homology theory. Cellular homology. Some examples. An introduction to higher homotopy groups.
- 8 October 2015: The Hopf Fibration. Cellular maps and induced homomorphisms in cellular homology.
13 October 2015: Two homotopic maps induce the same homomorphism
in cellular homology. The independence of cellular homology from the
- 15 October 2015: Pairs of Cw-complexes and subcomplexes. The long exact sequence in relative homology.
- 20 October 2015: Relative homology and its connection with reduced homology. The wedge of spaces and its homology.
- 22 October 2015: The Mayer-Vietoris exact sequence and its application to computation of homology groups.
- 27 October 2015: Singular homology and its connection to cellular homology, I.
- 29 October 2015: Singular homology and its connection to cellular homology, II.
- 3 November 2015: Singular cohomology. Ext and Tor functors.
- 10 November 2015: The universal coefficient theorem for cohomology. The cohomology of real projective space.
- 12 November 2015: The universal coefficient theorem for homology and the Kuenneth formula.
- 17 November 2015: Cup and Cap product. Examples and definitions.
- 24 November 2015: Orientability of manifolds and Poincaré duality.
- 26 November 2015: The proof of Poincaré duality.
- 1 December 2015: Some results on vector bundles I.
- 3 December 2015: Some results on vector bundles II.
- 10 December 2015: Overview and some problems.