Undecidability in Topology

The purpose of this lecture is to make explicit the limits of computational topology by showing that some simple and natural questions in topology are undecidable. In order to make the statement precise we need to define the notion of decidability and to specify the description of topological spaces we are interested in. Concerning topological spaces we should consider spaces having a combinatorial description such as finite simplicial complexes1. Note that many interesting spaces have such a description: compact topological manifolds of dimensions 2 or 3, compact differentiable manifolds, etc. See [Man14] for a survey. Concerning decidability there are essentially two notions. One refers to the independence of a statement with respect to a logical system. In other words, the statement is undecidable if neither its affirmation nor its negation can be proved from the axioms of the system using its logical rules. The existence of such undecidable statements relates to the first Gödel’s incompleteness theorem. The other notion of decidability refers to a family of problems with YES/NO answers, such as testing a property over a family of objects, and expresses the existence of an algorithm to output the answer of any problem in the family. Note that any finite family of problems for which the answers is provable is always decidable