Algebraic Topology, Master Degree, 6 cfu

Lectures: prof. Gilberto Bini

Goals

This course aims at introducing 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 and homotopy theory. Time permitting, we will introduce persistent homology and go over some of its recent applications to various areas.


Exams

Please find below a list of problems taken from the textbook [S]. You should return ten of them 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 syllabus.

Problems
  • Exercise 1.1.19: (iv), (vii), (v), (xii), (xiii);
  • 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);

  • References
  • [B] G. E. Bredon, Geometry and Topology, GTM Springer 139 , NY, 1993.
  • [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, 2014



  • Syllabus
  • 26 September 2016: General introduction. Homology of a complex. Singular homology.
  • 4 October 2016 (one hour): The boundary operator. Arcwise connected components and H0.
  • 6 October 2016: Review of the fundamental group and relation with the first homology group.
  • 11 October 2016: The homomorphism between homology group that is induced from continuous maps between topological space. Chain maps.
  • 13 October 2016: Topological pairs and relative homology. The long exact sequence in relative homology. The connecting homomoprhims.
  • 18 October 2016: Homology theory via the axioms of Eilenberg and Steenrod. The homology of spheres.
  • 20 October 2016: Applications of the homology of spheres. The definition of degree.
  • 25 October 2016: CW-complex of finite type. Applications and various examples.
  • 3 November 2016: Rational Homology Spheres.
  • 8 November 2016: Cellular Homology: first examples and statements.
  • 10 November 2016: The cellular homology complex. Singular homology is isomorphic to Cellular homology
  • 15 November 2016: Examples of cellular homology: closed and compact topological surfaces, complex projective space and real projective space
  • 17 November 2016: Some consequences of the generalized Jordan curve theorem. The invariance of dimension
  • 22 November 2016: Tensor products and Hom functor
  • 24 November 2016: The homology module with coefficients
  • 29 November 2016: The singular cohomology with G coefficients
  • 1 December 2016: The universal coeffcient theorem (for homology and cohomology theory)
  • 6 December 2016: Cup product. The cohomology ring. Examples. The cohomology ring of complex projective space and real projective space (with Z_2 coefficients)
  • 13 December 2016: A review of differential forms on differentiable manifolds. The de Rham cohomology Theorem. Poincaré duality
  • 15 December 2016: The proof of Poincaré duality
  • 20 December 2016: The de Rham Theorem. I
  • 10 January 2017: The de Rham Theorem. II

  • phone: +39-(0)2-50316180
    fax: +39-(0)2-50316090