Simply-connected, spineless 4-manifolds

TL;DR

This paper constructs infinitely many simply-connected smooth 4-manifolds homotopy equivalent to S^2 that lack a PL embedding of S^2, using Heegaard Floer invariants to identify obstructions.

Contribution

It provides the first examples of such 4-manifolds without spines, solving a key case in the existence problem for codimension-2 spines.

Findings

Existence of infinitely many such 4-manifolds.

Obstruction identified via Heegaard Floer d-invariants.

No PL embedding of S^2 exists in these manifolds.

Abstract

We construct infinitely many smooth 4-manifolds which are homotopy equivalent to $S^2$ but do not admit a spine, i.e., a piecewise-linear embedding of $S^2$ which realizes the homotopy equivalence. This is the remaining case in the existence problem for codimension-2 spines in simply-connected manifolds. The obstruction comes from the Heegaard Floer $d$ invariants.