Finite topological spaces and some classical conjectures in algebraic and geometric topology

MINIAN, ELIAS GABRIEL

Cordoba

Congreso; IV CLAM; 2012

At first sight, a non-Hausdorff topological space with finitely many points might seem uninteresting. However in this talk I will show that the theory of finite spaces can be used to study deep classical problems in topology. In the first part of my exposition I will recall the basic ideas, results and examples on the homotopy theory of finite spaces, including its relationship with the homotopy theory of polyhedra. We will see the foundational results of Stong and McCord and some of the most important results that we obtained over the last years. In the second part of the lecture we will discuss how to use the theory of finite spaces to attack two open problems in topology: the Andrews-Curtis conjecture, and Whitehead's asphericity question.