Computability of finite simplicial complexes

Abstract

The topological properties of a set have a strong impact on its computability properties. A striking illustration of this idea is given by spheres and closed manifolds: if a set $X$ is homeomorphic to a sphere or a closed manifold, 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, Iljazovi\'c, Su\v{s}i\'c and others in the recent years. A similar notion of computable type was also defined for pairs $(X,A)$ in order to cover more spaces, such as compact manifolds with boundary and finite graphs with endpoints. We investigate the higher dimensional analog of graphs, namely the pairs $(X,A)$ where $X$ is a finite simplicial complex and $A$ is a subcomplex of $X$. We give two topological characterizations of the pairs having computable type. The first one uses a global property of the pair, that we call the $\epsilon$-surjection property. The second one uses a local property of neighborhoods of vertices, called the surjection property. We give a further characterization for $2$-dimensional simplicial complexes, by identifying which local neighborhoods have the surjection property. Using these characterizations, we give non-trivial applications to two famous sets: we prove that the dunce hat does not have computable type whereas Bing's house does. Important concepts from topology, such as absolute neighborhood retracts and topological cones, play a key role in our proofs.