Four manifolds with no smooth spines (Q2171421)

A spine is a (not necessarily locally flat) submanifold that is a strong deformation retract of the ambient manifold. Depending on the kind of submanifold, the spine is topological, smooth or PL. There exist examples of 4-manifolds that are homotopy equivalent to a closed surfaces but do not admit a topological spine [\textit{G. A. Venema}, Topology Appl. 90, No. 1--3, 197--210 (1998; Zbl 0930.57022)], and there exist 4-manifolds that have a topological spine but do not admit a PL spine [\textit{H. J. Kim} and \textit{D. Ruberman}, Algebr. Geom. Topol. 20, No. 7, 3589--3606 (2020; Zbl 07335436)]. The authors use obstructions from Heegard Floer theory to construct examples of 4-manifolds with PL spines that admit no smooth spine. More precisely, they show the following. Let \(S\) be a spine in a compact oriented smooth 4-manifold. If \(S\) is locally flat, then \(S\) is smoothable by Corollary 6.8 of [\textit{C. P. Rourke} and \textit{B. J. Sanderson}, Ann. Math. (2) 87, 1--28 (1968; Zbl 0215.52204)]. One can always arrange \(S\) to have at most one non-locally-flat point. If the singularity knot \(K\) for this point is smoothly slice, then again \(W\) has a smooth spine. In contrast, the authors show that if \(K\) \begin{itemize} \item has nonzero Arf invariant, or \item is a nontrivial L-space knot, or \item is the nontrivial connected sum of nontrivial L-space knots, or \item is an alternating knot of signature \(<-4\), \end{itemize} then \(W\) contains no smooth spine. The authors also construct for each \(g,e\in \mathbb Z\) with \(g\geq 0\) a compact smooth oriented 4-manifold that has no smooth spine but admits a topological locally flat spine that is an oriented closed genus \(g\) surface with normal Euler number \(e\).