Four manifolds with no smooth spines

TL;DR

This paper investigates conditions under which certain 4-manifolds cannot be deformed onto smooth surfaces, using advanced topological invariants and surgery theory to identify obstructions related to knot properties.

Contribution

It introduces new obstructions based on Heegaard Floer homology and surgery theory that prevent some 4-manifolds from having smooth spines, especially linked to specific knot invariants.

Findings

4-manifolds with nonzero Arf invariant knots lack smooth spines.

L-space knots and their connected sums also obstruct smooth spines.

Alternating knots with signature less than -4 prevent smooth spines.

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.