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Abstract

Let $W$ be a compact smooth $4$-manifold that deformation retract to a PL
embedded closed surface. One can arrange the embedding to have at most one
non-locally-flat point, and near the point the topology of the embedding is
encoded in the singularity knot $K$. If $K$ is slice, then $W$ has a smooth
spine, i.e., deformation retracts onto a smoothly embedded surface. Using the
obstructions from the Heegaard Floer homology and the high-dimensional surgery
theory, we show that $W$ has no smooth spines if $K$ is a knot with nonzero Arf
invariant, a nontrivial L-space knot, the connected sum of nontrivial L-space
knots, or an alternating knot of signature $<-4$. We also discuss examples
where the interior of $W$ is negatively curved.