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 relations. For these purposes, we will recall some fundamental topics, such as CW-complexes and homotopy theory. Time permitting, we will introduce persistent homology and go over some of its recent applications to various areas, such as biomedical engineering.


Exams

Please download here the worksheet for the exam. You should return it at most one week before you are going to take the oral exam. In addition to some questions about the worksheet, the exam will go over the material covered in the syllabus.
  • 27 September 2017. The homology of a complex. The singular homology of a topological space
  • 28 Septembre 2017. The 0-th homology group and the number of connected components of a topological space
  • 4 October 2017. The first homology group and the fundamental group
  • 5 October 2017. The Eilenberg-Steenrod System of axioms for homology. An introduction to the singular cohomology
  • 11 October 2017. Homotopic equivalent maps induce the same homomorphism in homology. Chain homotopic maps between complexes
  • 12 October 2017. The connecting homomorphism and applications to relative homology theory
  • 18 October 2017. The Mayer-Vietoris exact sequence and applications to spheres, complex projective space, etc.
  • 19 October 2017. Degree of continuous maps and applications to vector fields
  • 25 October 2017. Reduced homology. Some properties of good pairs in relative homology.
  • 26 October 2017 (1 hour). Finite CW-complexes. Examples and applications.
  • 2 November 2017. The cellular complex and the cellular homology.
  • 8 November 2017. Some calculations in cellular homology.
  • 9 November 2017. The PoincarĂ© dodecahedral space.
  • 15 November 2017. No lecture.
  • 16 November 2017. The tensor and the tor functor. The universal coefficient theorem.
  • 22 November 2017. The Hom and the Ext functor. The universal coefficient theorem for cohomology.
  • 23 November 2017. Some applications of the universal coefficient theorem.
  • 29 November 2017. Cup and cap product.
  • 30 November 2017. No lecture.
  • 06 December 2017. The structure of the cohomology ring. Some examples.
  • 13 December 2017. The orientability of manifolds and PoincarĂ© duality.
  • 14 December 2017. Introduction to persistent homology.
  • 20 December 2017. Some examples in persistent homology.
  • 21 December 2017. Some applications of persistent homology.


  • 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

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