Simply Connected, Spineless 4-Manifolds

Abstract

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.

The trace embedding lemma and spinelessness

“…This provides an alternative solution to Problem 4.25 in Kirby's list [34], which was recently resolved by Levine and Lidman [38]. Levine and Lidman produced examples of smooth, compact 4-manifolds that are homotopy equivalent to S 2 but do not contain S 2 as a PL spine, as detected using the d-invariants in Heegaard Floer homology.…”

“…Levine and Lidman produced examples of smooth, compact 4-manifolds that are homotopy equivalent to S 2 but do not contain S 2 as a PL spine, as detected using the d-invariants in Heegaard Floer homology. (Subsequently Kim and Ruberman used surgery theory to show that infinitely many of the examples from [38] contain S 2 as a topological spine with cone points.) In fact, the argument used by Levine and Lidman shows that no smooth 4-dimensional homotopy 2-sphere with the same boundary and intersection form as the examples from [38] can contain S 2 as a PL spine.…”

“…(Subsequently Kim and Ruberman used surgery theory to show that infinitely many of the examples from [38] contain S 2 as a topological spine with cone points.) In fact, the argument used by Levine and Lidman shows that no smooth 4-dimensional homotopy 2-sphere with the same boundary and intersection form as the examples from [38] can contain S 2 as a PL spine. In contrast, our results show that the existence of spines depends on the smooth structure; each of our spineless 4-manifolds is homeomorphic to a 4-manifold that admits a smooth spine.…”

“…Among manifolds with boundary, work of Gordon [25] implies that a contractible 4-manifold with non-simply-connected boundary is never geometrically simply connected; see also [33, p.8] and [8]. Levine and Lidman's results show that there are 3-manifolds Y such that every smooth 4-dimensional homotopy 2-sphere with boundary Y and positive intersection form requires 1-handles [38,Remark 1.2]. We give the first examples demonstrating that, as expected, geometric simple-connectivity can depend on the smooth structure: Corollary A.2 There exist infinitely many pairs of compact, homeomorphic 4-manifolds W and W such that W is geometrically simply connected but W is not.…”

“…In order to obstruct piecewiselinear embeddings, the standard trick is to show that there is some set of knot invariants which have to take on a special set of values for any knot arising as the link of the singularity, and then show that there is no knot in S 3 realizing the entire set of invariants. (For a selection of applications of this strategy to various problems, see [20] on almost concordance, [29] on homology concordance, [3] on rational cuspidal curves, and [38] on spinelessness.) This clever strategy yields powerful obstructions but is often computationally demanding.…”