  A 2018 paper by A. Levine and T. Lidman outlines a proof of the following
interesting result in topology of manifolds: there is a compact smooth
4-manifold $W$ with boundary such that $W$ is homotopy equivalent to $S^2$ but
there does not exist an embedding $S^2\to W$ which is a homotopy equivalence
and is simplicial for some triangulations of $W$ and of $S^2$.
  We present a shorter (and hopefully clearer) exposition. We reveal that some
parts of the proof are missing, and that some results are used in that paper
without proof or reference, or even without explicit statement.
