Four manifolds with no smooth spines

Let $W$ be a compact smooth orientable $4$-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 spine 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 $\lt-4$. We also discuss examples where the interior of $W$ is negatively curved.

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Belegradek was partially supported by the Simons Foundation grant 524838.

Received 5 March 2021

Received revised 24 August 2021

Accepted 13 September 2021

Published 6 September 2022