Knot traces and PL embeddings

It remains at the forefront of 4-manifold topology to construct simple closed 4-manifolds with distinct smooth structures. Towards that end, it is of interest to construct simple 4-manifolds with boundary with very distinct smooth structures. For all genera g, we produce pairs of homeomorphic smooth 4-manifolds Z and Z’ which are homotopy equivalent to a genus g surface and which have smooth structures distinguished by several formal properties: Z is diffeomorphic to a genus g knot trace and Z’ is not, Z admits a smooth spine and Z’ does not admit a piecewise linear spine. When g = 0, Z is geometrically simply connected and Z’ is not. In particular our Z’ are simple spineless 4-manifolds, which gives an alternative to Levine and Lidman’s recent solution to Problem 4.25 on Kirby’s list. This is joint work in progress with Kyle Hayden.