We present a 4-dimensional manifold which is homeomorphic (but non-diffeomorphic) to the 4-sphere punctured along a Cantor set. It is shown that this manifold is not diffeomorphic to any leaf of any C2 codimension 1 foliation on a compact manifold, the proof involves a mixing of topological and dynamical tools: Donaldson's and Furuta's theorems, Reeb stability, Novikov's Theorem or Kopell's Lemma.
This result is also true in greater generality: the 4-sphere can be replaced by any simply connected closed 4-manifold with trivial Kirby-Siebenmann invariant and the Cantor subset by any compact totally disconnected subset. Moreover, there are a continuum of such examples (by Taubes' Theorem). This is a joint work with P. A. Schweitzer (PUC-Rio) and can be considered as a natural continuation of our previous work: https://arxiv.org/abs/1410.8182v7. The main goal is to get similar results in regularity C1.
© Carlos Meniño Cotón.