Jonathan Barmak: Mini course on "Finite topological spaces"

Although finite metric spaces are discrete, a topological space with finitely many points need not be discrete. In fact, the homotopy theory of finite spaces is as rich as the homotopy theory of complexes. For every compact polyhedron there exists a finite space with the same homology and homotopy groups.

We will study the correspondence between finite spaces and partially ordered sets and we will see that homotopies and homotopy types of finite spaces are easy to describe in a combinatorial way. We will analyze the relationship between finite spaces and simplicial complexes at three different levels: homotopy, weak homotopy and simple homotopy.

We will see how two open conjectures in discrete geometry and group theory can be reinterpreted from the optic of finite spaces.

The course will be almost self-contained. Only basic notions of topology and algebraic topology will be required and most of these will be recalled during the lectures. The proofs and ideas will be easy to follow for graduate and undergraduate students.

The course will consist of four one-hour lectures.

Dates: Wednesdays 27/4, 4/5, 11/5 and 18/5.

Time: 9:00 - 10:00.