Topologies on finite sets, homotopies and fixed points

BARMAK, JONATHAN ARIEL

Bedlewo

Workshop; Winter Workshop on Dynamics, Topology and Computations; 2018

FACULTY OF MATHEMATICS AND COMPUTER SCIENCE JAGIELLONIAN UNIVERSITY IN KRAKÓW, STEFAN BANACH INTERNATIONAL MATHEMATICAL CENTER

Finite metric spaces are useful when studying infinite spaces, for instance in Data Analysis. However the topology of a finite metric space is not interesting: two such spaces with the same cardinality are necessarily homeomorphic (indistinguishable from a topological viewpoint). A topological space with finitely many points is a much richer object. Finite spaces can be used to model well-known Hausdorff spaces, such as manifolds or polyhedra. Given any triangulated space, there is a finite space with the same homology and homotopy groups. In contrast to simplicial complexes, there exists an algorithm due to Stong which decides whether two finite spaces can be deformed one into the other (homotopy equivalence). These ideas are used to prove that an action of a group G on a contractible finite T0 space always has a fixed point. This is not true for contractible compact polyhedra. The homotopy theory of finite spaces can be used to study a conjecture by Quillen about the poset of p-subgroups of a group G.In this course we will see how finite spaces can be studied combinatorially using posets, and we will present the results of McCord which relate finite spaces and polyhedra. We will study homotopy types of finite spaces and establish connections between homotopy and fixed point properties.