On the difficulty of classifying open manifolds

Descriptive set theorists have developed a hierarchy of levels of unclassifiability for uncountable collections of various mathematical objects. They are now applying it to other fields of mathematics. We will look at various applications to manifold topology. Contractible 3-manifolds and exotic smoothings of R^4 cannot be classified (up to diffeomorphism) by any countable collection of numerical invariants. Other manifold classification problems in dimensions 2 and above reach much higher levels in the hierarchy. For example, this applies to the diffeomorphism classification of smoothings on certain open manifolds of dimension 4 and higher, to the proper homotopy classification within certain homotopy types in every dimension at least 3, and to the homeomorphism classification of certain proper homotopy types. This is work in progress with Aristotelis Panagiotopoulos.