Computability of Compact Spaces
Time: 14:00 -- Location: LRI, 445
summary: The topological properties of a set have a strong impact on its computability properties. A striking illustration of this idea is given by spheres, closed manifolds and finite graphs without endpoints : if a set X is homeomorphic to a sphere, a closed manifold or such a graph, then any algorithm that semicomputes X in some sense can be converted into an algorithm that fully computes X. In other words, the topological properties of X enable one to derive full information about X from partial information about X. In that case, we say that X has computable type. Those results have been obtained by Miller and Iljazović and others in recent years. We give a characterization of finite simplicial complexes having computable type using a local property of stars of vertices. We argue that the stronger, relativized version of computable type, is better behaved. Consequently, we obtain characterizations of strong computable type, related to the descriptive complexity of topological invariants. We study two families of topological invariants of low descriptive complexity, expressing the extensibility and the null-homotopy of continuous functions. We apply the theory to revisit previous results and obtain new ones. This leads us to investigate the expressive power of low complexity invariants in the Borel hierarchy and their ability to differentiate between spaces. Notably, we show that they are sufficient for distinguishing finite topological graphs.