Piecewise-linear disks and spheres in 4-manifolds

Adam Levine (Duke)

I will discuss a variety of problems related to the existence of piecewise-linear (PL) embedded surfaces in smooth 4-manifolds (joint work with Jen Hom and Tye Lidman). We give the first known example of a smooth, compact 4-manifold which is homotopy equivalent to the 2-sphere but for which the homotopy equivalence cannot be realized by a PL embedding. We also show that the PL concordance group of knots in homology 3-spheres is infinitely generated and contains elements of infinite order.