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MAI0126 Topological combinatorics


This course is divided into two parts. Approximately six lectures will be devoted to each part. During the first, we shall review some classical applications of topology to combinatorial problems. A highlight is the idea, pioneered by Lovász in the seventies, of using the Borsuk-Ulam theorem for proving nonexistence of graph colourings. This beautiful approach can be seen as the historical starting point for the topological method in combinatorics.

In the second part we shall study combinatorial methods for computing topological invariants of simplicial complexes and more general cell complexes. The development of these methods has often been driven by the desire to compute invariants of specific complexes that were inaccessible by previously known methods. One goal of the course is to gain familiarity with these particularly interesting concrete constructions.

Course content

Versions of the Borsuk-Ulam theorem with applications to combinatorics. Theory of simplicial and cellular complexes. Combinatorial methods (shellability, discrete Morse theory, nerve lemmas, fiber lemmas, etc.) for computing topological invariants. Classical examples of combinatorially defined cell complexes and their topological properties.


For the first part of the course, we shall mainly use:
J. Matoušek, Using the Borsuk-Ulam Theorem, Springer, 2003.

The second part is based on various handouts and/or freely downloadable material.


Basic abstract algebra and some discrete mathematics. It is recommended to have taken a course in topology, but it is not a strict requirement; along the way, we shall briefly review the material we need.


Homework assignments and a literature project where each participant is expected to give a brief seminar on some topic related to the course.


Axel Hultman

Sidansvarig: axel.hultman@liu.se
Senast uppdaterad: 2019-11-29