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Heegaard Floer homology and the word metric on the Torelli group

Santana Afton, Miriam Kuzbary, Tye Lidman

TL;DR

This work investigates how Heegaard Floer $d$-invariants of integral homology spheres behave under Torelli surgery, reframing the problem in terms of the word metric on the Torelli group generated by separating twists and bounding pair maps. It proves a linear-type bound on the $d$-invariant change in terms of Torelli word length, and uses this to show the Cayley graph has infinite diameter while identifying broad classes of Torelli surgeries that keep $d$-invariants bounded. The paper also establishes that no Morita-type formula extends to rational homology spheres associated with level-$d$ congruence subgroups, providing explicit counterexamples. Overall, it connects 3-manifold invariants with algebraic properties of the Torelli group, highlighting both rigid and flexible aspects of Torelli surgery in different settings.

Abstract

We study a relationship between the Heegaard Floer homology correction terms of integral homology spheres and the word metric on the Torelli group. For example, we give an elementary proof that the Cayley graph of the Torelli group has infinite diameter in the word metric induced by the generating set of all separating twists and bounding pair maps. On the other hand, we show that many subsets of the Torelli group are bounded with respect to this metric. Finally, we address the case of rational homology spheres by ruling out a certain Morita-type formula for congruence subgroups of mapping class groups.

Heegaard Floer homology and the word metric on the Torelli group

TL;DR

This work investigates how Heegaard Floer -invariants of integral homology spheres behave under Torelli surgery, reframing the problem in terms of the word metric on the Torelli group generated by separating twists and bounding pair maps. It proves a linear-type bound on the -invariant change in terms of Torelli word length, and uses this to show the Cayley graph has infinite diameter while identifying broad classes of Torelli surgeries that keep -invariants bounded. The paper also establishes that no Morita-type formula extends to rational homology spheres associated with level- congruence subgroups, providing explicit counterexamples. Overall, it connects 3-manifold invariants with algebraic properties of the Torelli group, highlighting both rigid and flexible aspects of Torelli surgery in different settings.

Abstract

We study a relationship between the Heegaard Floer homology correction terms of integral homology spheres and the word metric on the Torelli group. For example, we give an elementary proof that the Cayley graph of the Torelli group has infinite diameter in the word metric induced by the generating set of all separating twists and bounding pair maps. On the other hand, we show that many subsets of the Torelli group are bounded with respect to this metric. Finally, we address the case of rational homology spheres by ruling out a certain Morita-type formula for congruence subgroups of mapping class groups.
Paper Structure (13 sections, 24 theorems, 32 equations, 7 figures)

This paper contains 13 sections, 24 theorems, 32 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgments
  4. Background
  5. The Torelli group
  6. Dehn twists and Dehn surgeries
  7. 3-manifold invariants
  8. Proofs of asymptotic results
  9. Families of homology spheres with bounded d-invariants
  10. Separating Dehn twists
  11. Bounding pair maps
  12. Combining separating twists and bounding pair maps
  13. Rational Homology Spheres

Key Result

Theorem 1.1

Let $S$ be an embedded surface of genus $g$ in a homology sphere $Y$. Let $A$ be a generating set for $\mathcal{I}_g$ consisting of bounding pair maps and separating twists. Let $k_A = \max_{\phi \in A} k_\phi$. Note that a generating set for $\mathcal{I}_g$ can always be chosen such that $k_A = 1$. where $||\phi||_A$ is the word length of $\phi$ in $\mathcal{I}_g$ and $d$ is the Heegaard Floer d-

Figures (7)

  • Figure 1: In this local picture of a curve $\gamma$ (in purple) lying on a surface $\Sigma$ (in green) embedded in a 3-manifold Y, we see the blue annulus is a neighborhood of the curve in the surface $\Sigma$, and the black solid torus is a neighborhood of $\gamma$ in $Y$. The meridian $\mu$ of $\gamma$ viewed as a knot in $Y$ is shown in orange. The result of cutting open $Y$ along $\Sigma$, performing a single positive Dehn twist on $\gamma \subseteq \Sigma$, and gluing back together along $\Sigma$ is exactly the same as removing the solid torus neighborhood of $\gamma$ and gluing it back with its meridian now going to the $\mu - \lambda$ curve shown on the right. This is precisely $-1$-surgery on $\gamma \subseteq Y$.
  • Figure 2: Above we see three equivalent surgery diagrams for the result of cutting open $S^3$ along a surface and twisting by $\phi^n$ where $\phi$ is a single Dehn twist along a simple closed curve in the surface isotopic to the knot $K$. In these diagrams, $s$ denotes the surface framing of the curve.
  • Figure 3: A schematic picture one dimension lower describing the proof of Lemma \ref{['lem:framinglemma']}.
  • Figure 4: An example of a curve $C_k$ in the proof of Lemma \ref{['lem:unbounded']} with $k=1$.
  • Figure 5: On the left is the twist knot $K_1$ as a separating curve, together with the subsurface it bounds on $\Sigma_2$. If we conjugate the Dehn twist $T_{K_1}$ by Dehn twists on the meridional curve on the right, we obtain the Dehn twist $T_{K_n}$ which is also a separating twist.
  • ...and 2 more figures

Theorems & Definitions (53)

  • Theorem 1.1
  • Corollary 1.2
  • proof
  • Remark 1.3
  • Corollary 1.3
  • proof
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • ...and 43 more