Fraisse theory has become one of the prominent frameworks for analyzing the dynamics of automorphism groups of countable structures. This method of analysis was later extended by Irwin and Solecki to homeomorphism groups Homeo(K) of compact metrizable spaces K. However, when it comes to applications, all examples which have been considered so far involve spaces K of dimension at most 1. In this talk I will survey some of the combinatorial techniques which are necessary for expanding the Fraisse-theoretic analysis to spaces of higher dimension.
After I review the basic Fraisse-theoretic background, I will define several categories of finite simplicial maps whose projective Fraisse limit is the same object: the "generic combinatorial simplex." In the process I will present some new results in the combinatorial counterpart to PL-topology which is known as the theory of combinatorial (stellar) manifolds.
This is joint work with Slawek Solecki.