A parametrised sumstable smoothing theorem for topological 4manifolds
A parametrised sumstable smoothing theorem for topological 4manifolds

Sander Kupers, University of Toronto
Fine Hall 314
The sumstable smoothing theorem of Freedman and Quinn implies that if the topological tangent manifold of a topological 4manifold admits a refinement to a vector bundle, then the 4manifold admits a smooth structure after finitely many connected sums with S^2 x S^2. We generalise this to families of topological 4manifolds, though in a weaker form than higherdimensional smoothing theory: we prove that a space of smooth structures, stabilised at varying locations, is homology equivalent to a space of vector bundle refinements of the topological tangent bundle, stabilised at varying locations. I will explain the statement, proof strategy, and some interesting consequences. This is joint work with Christian Kremer.