Knot traces and PL embeddings
It is still at the forefront of 4-manifold topology to produce (algebraically) small 4-manifolds which admit exotic smooth structures. Knot traces are 4-manifolds built by attaching a (thickened) disk to B^4; these fundamental manifolds are smooth and homotopy equivalent to S^2. We build smooth 4-manifolds which are homeomorphic to a knot trace but not diffeomorphic to any knot trace; this shows that knot traces can even admit an exotic smooth structure with fundamentally distinct Morse theory. As a corollary, we show that there are smooth 4-manifolds homotopy equivalent to a genus g surface such that there is no PL embedding of the surface that realizes the homotopy equivalence. This gives a new proof of Lidman-Levine's recent solution to problem 4.25 on Kirby's list. This is joint work in progress with Kyle Hayden.