Whitney tower concordance and knots in homology spheres

TL;DR

This paper shows that every link in a homology sphere is height-$h$ Whitney tower concordant to some link in $S^3$ for any $h$, indicating that the symmetric Whitney tower filtration collapses homology-sphere links to $S^3$ at this level. It clarifies the relationship between symmetric Whitney towers, relative towers, and the solvable concordance filtration, proving that height-$h$ Whitney tower concordance implies $h$-solvable concordance and thus recovering Davis's result that the solvable filtration cannot distinguish links in homology spheres from those in $S^3$. The work strengthens the bridge between topological methods (Whitney towers, relative towers) and concordance obstructions, highlighting a topological equivalence that contrasts with known smooth obstructions. Overall, the results emphasize the power of Whitney tower techniques in understanding link concordance in homology spheres and their connection to classical filtrations.

Abstract

In a groundbreaking work A. Levine proved the surprising result that there exist knots in homology spheres which are not smoothly concordant to any knot in $S^3$, even if one allows for concordances in homology cobordisms. Since then subsequent works due to Hom-Levine-Lidman and Zhou have strengthened this result showing that there are many knots in homology spheres which are not smoothly concordant to knots in $S^3$. In this paper we present evidence that the opposite is true topologically. We study the Whitney tower filtration of concordance due to Cochran-Orr-Teichner and prove that modulo any term in this filtration every knot (or link) in a homology sphere is equivalent to a knot (or link) in $S^3$. As an application we recover the main result of [Davis2019], namely that the solvable filtration similarly fails to distinguish links in homology spheres from links in $S^3$.

Figures (10)

• Figure 1: Left to right: A Whitney disk pairing up two points of intersection. A (portion of a) Whitney tower. The Whitney moves canceling two points of intersection.
• Figure 2: Left: An immersed union of annuli $A$ bounded by $K\subseteq Y$ and $U\subseteq S^3$ extending to a height $1$ Whitney tower. Middle: Capping $A$ with disks bounded by $U$ produces an immersed union of disks extending to a height $1$ Whitney tower. Right: A immersed union of disks $D_1\cup D_2$ bounded by a link $K$ together with a pair of small 4-balls $B_1$ and $B_2$. Tube $B_1$ and $B_2$ together to get a single 4-ball $B$. Remove the interior of $B$ to arrive at a Whitney tower concordance from $K$ to the unlink.
• Figure 3: Left: A double point $p$ in an immersed surface together with arcs $\alpha_1$ and $\alpha_2$ running from $p$ to points in $\partial W \cap S$ and a third arc $\alpha_3$ connecting the endpoints of $\alpha_1$ and $\alpha_2$. Center: $\alpha_1*\alpha_3*\overline{\alpha_2}$ bounding an immersed disk, called a relative Whitney disk. Right: The relative Whitney trick along $\Delta$ removes the double point at $p$.
• Figure 4: Left: Two arcs in $S\cap \partial W$ connected by the relative Whitney arc $\alpha_3$ of a relative Whitney disk. Center: The relative Whitney move changes $S\cap \partial W$ by sliding along $\alpha_3$ over a meridian. Right: The partial relative Whitney trick introduces a new point of intersection and replaces a relative Whitney disk with a Whitney disk.
• Figure 5: Left: An arc $\beta\subseteq W$ between two points in immersed surface $S\subseteq W$. Right: The result of the finger move $\beta$ along with an embedded Whitney disk which undoes the finger move.

Theorems & Definitions (22)

• Conjecture 1.1
• Theorem 1.2
• Corollary 1.3: Theorem 1.2 of Davis2019
• Definition 2.1: Definition 7.7 of COT03, Definition 2.5 of Cha2014
• Remark 2.2
• Remark 2.3
• Definition 2.4
• Definition 2.5: Compare with Definition 2.12 of Cha2014.
• Proposition 2.6
• proof