Speaker: Michał Kukieła (UMK)

Title of the talk: Some combinatorial approaches to homotopy

Abstract: Different notions of “homotopy” equivalences of partially ordered sets may be defined in terms of various one-point reductions and expansions. These have been a recent object of study of J.A. Barmak and G.E. Minian. Their research was inspired by results from the 60's of R.E. Stong, who classified, using elementary “deformations”, the homotopy types of finite topological spaces.
Finite spaces satisfying the T0 separability axiom may be easily identified with partially ordered sets, and the deformations of Stong turn out to be dismantlings by irreducible points. Some, natural from a topologist's point of view, generalizations of irreducible points give interesting definitions of “homotopy”.
I will present relations between these notions and their connections to topics such as poset fixed point theory, evasiveness and homotopy theory of polyhedra.