96769 - Computational Topology

Academic Year 2023/2024

  • Teaching Mode: Traditional lectures
  • Campus: Bologna
  • Corso: Second cycle degree programme (LM) in Mathematics (cod. 5827)

Learning outcomes

At the end of the course, the student knows some topics in computational topology, such as methods to study the shape of point clouds via the construction of suitable simplicial complexes (e.g., Čech and Vietoris-Rips complexes) and the use of simplicial homology and cohomology.

Course contents

Affine and convex combinations and hulls. Simplices. Geometric simplicial complexes. Finite abstract simplicial complexes. Isomorphisms between finite abstract simplicial complexes. Vertex schemes and geometric realizations. Geometric realization theorem for finite abstract simplicial complexes.

Euler characteristic.

Barycentric coordinates. Vertex maps, simplicial maps, PL maps.

Subdivisions of geometric simplicial complexes. Barycentric subdivisions. Mesh lemma and its corollary.

Simplicial approximations. Lebesgue’s number lemma. Simplex lemma. Simplicial approximation theorem. Simplicial approximation theorem in metric form.

Nerve and nerve theorem. Čech complexes and Vietoris-Rips complexes.

Concept of miniball. Existence and uniqueness of the miniball of a compact set. Vietoris-Rips lemma and its optimality. Voronoi diagram. Delaunay complex and its main properties. Alpha-complexes and their main properties.

Group of p-chains. Boundary operator and fundamental lemma of homology. Chain complexes. Homology groups of a chain complex. Computation of Betti numbers. Computation of bases for cycle groups, boundary groups and homology groups.

Euler-Poincaré theorem.

Reduced homology.

Chain maps and induced homomorphisms between homology groups. Chain maps induced by simplicial applications.

Definition of category. Functors. The homology functor. Collapsibility. Invariance of homology groups under elementary collapses.

Relative homology. Computation of relative homology groups. Excision theorem.

Exact sequences of chain complexes and chain maps. Snake Lemma. Exact sequence of a pair and Mayer-Vietoris sequence as consequences of the Snake Lemma. Reduced homology of S^n.

Dual of a vector space and dual of a homomorphism between vector spaces. Cochains. Algebraic coboundary operator. Cocycles, coboundaries. Cohomology groups. Geometric coboundary operator. Reduced simplicial cohomology. Computation of simplicial cohomology through the ranks of the coboundary matrices. Statement of Poincaré's duality theorem. Euler characteristic and Poincaré's duality. Lefschetz’s duality. Reeb graphs.

Topological groups. If X is a compact metric space, then Homeo(X) is a topological group that acts continuously on C^0 (X, R). Definition of the natural pseudo-distance associated with a subgroup G of Homeo (X). Main properties of the natural pseudo-distance with respect to the group Homeo(X). Notes on singular homology with coefficients in Z_2.

Links between the natural pseudo-distance, persistent homology, and the theory of non-expansive equivariant operators.


H. Edelsbrunner and J.L. Harer, Computational topology: An introduction, American Mathematical Society, 2010.

Teaching methods

Lecture of traditional type.

Assessment methods

Written exercises and oral examination.

Teaching tools

See the web page http://www.dm.unibo.it/~frosini/DIDMAT.shtml

Links to further information


Office hours

See the website of Patrizio Frosini