Four manifolds with no 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.