SIMPLY CONNECTED, SPINELESS 4-MANIFOLDS

We construct infinitely many compact, smooth 4-manifolds which are homotopy equivalent to$S^{2}$but do not admit a spine (that is, a piecewise linear embedding of$S^{2}$that 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.