Genus two Goeritz equivalence in lens spaces $L(p,1)$

Abstract: In this paper, we consider the action of the Goeritz group $\mathcal G_p$ for the genus two Heegaard splitting of the lens space $L(p,1)$ with $p\ge 2$ on the homology of the Heegaard surface. We describe the action in terms of matrices in $GL(4, \mathbb Z)$, and provide homology and homotopy obstructions for when two curves in the Heegaard surface are Goeritz equivalent.

• The paper computes the star representation of the genus two Goeritz group of L(p,1), identifies its kernel as the normal subgroup generated by the belt-curve twist β², and characterizes its image, including restrictive inequalities for p ≥ 5.
• The authors derive arithmetic homology obstructions—covering congruences, gcd invariance, divisibility, and inequalities—that provide a practical sieve for testing whether two curves can be Goeritz equivalent, especially in lens space surgery problems.
• The paper proves a homotopy obstruction: homologous curves that are not freely homotopic in either handlebody cannot be related by a Goeritz automorphism, while noting that the criteria are necessary rather than sufficient and are limited to L(p,1).

This paper by Doleshal and Rathbun extends a program begun in their earlier work on the genus two Goeritz group of $S^3$ to lens spaces of the form $L(p,1)$ for $p \geq 2$. The Goeritz group $\mathcal G_p$ here is the group of isotopy classes of orientation-preserving self-homeomorphisms of $L(p,1)$ preserving the (unique) genus two Heegaard splitting set-wise. The authors' goal is to give computable obstructions to when two curves on the genus two Heegaard surface are related by such an automorphism — a question motivated by Berge's conjecture and by known examples of knots admitting inequivalent primitive positions on a genus two Heegaard splitting yet inducing identical surgery slopes (2602.16458).

The star map on homology

The paper builds directly on Cho's presentations for $\mathcal G_p$, which distinguish three regimes: $p = 2$ (generated by $\beta$, $\rho$, $\gamma$), $L(p,1)$0 (a direct summand generated by $L(p,1)$1 plus $L(p,1)$2), and $L(p,1)$3 ($L(p,1)$4). Fixing a basis $L(p,1)$5 for $L(p,1)$6 adapted to the handlebody $L(p,1)$7, the authors compute the induced matrices for every generator, producing a representation

$L(p,1)$8

with explicit $L(p,1)$9 integral matrices $p \geq 2$0. The computations are carried out by tracking oriented intersection numbers of generator images against the basis curves, using pants decompositions (for $p \geq 2$1) and pillowcase decompositions (for $p \geq 2$2) induced by primitive disk pairs. A notable subtlety arises in defining $p \geq 2$3: the relative orientations of the pairs $p \geq 2$4 and $p \geq 2$5 are not immediately determined by the geometry, and the authors resolve the ambiguity via a sign argument on intersections with the disk $p \geq 2$6.

Structure of the image and kernel

The central structural result is that $p \geq 2$7 lies in an explicitly described subgroup $p \geq 2$8 of $p \geq 2$9 consisting of block matrices built from $\mathcal G_p$0 unimodular matrices in $\mathcal G_p$1 of the form $\mathcal G_p$2. The projection $\mathcal G_p$3 onto the upper-left block is shown to be injective, which reduces all subsequent analysis to $\mathcal G_p$4 matrix arithmetic.

The kernel is identified precisely: the kernel of the star map is exactly the normal subgroup generated by $\mathcal G_p$5, the Dehn twist about the belt curve separating $\mathcal G_p$6 from $\mathcal G_p$7. This is proven case-by-case via normal forms for words in $\mathcal G_p$8: sign/positivity arguments handle $\mathcal G_p$9 and $L(p,1)$0, while for $L(p,1)$1 injectivity follows from freeness of the group generated by the Möbius transformations of $L(p,1)$2 and $L(p,1)$3, using Lyndon–Ullman.

The image is then characterized completely:

Range	Image of star map
$L(p,1)$4	All of $L(p,1)$5
$L(p,1)$6	Matrices from $L(p,1)$7, i.e., those additionally satisfying $L(p,1)$8 or $L(p,1)$9

The inequality constraint becomes genuinely restrictive only at $\mathcal G_p$0; for $\mathcal G_p$1 it is vacuous because it would contradict $\mathcal G_p$2. This is the sharpest arithmetic statement in the paper: for large $\mathcal G_p$3, the Goeritz action on homology is a proper subgroup of the natural candidate group, so some homologically realizable maps are not geometrically realized.

Homology obstructions

Suppose $\mathcal G_p$4 for homology vectors $\mathcal G_p$5, $\mathcal G_p$6 of curves $\mathcal G_p$7 on $\mathcal G_p$8. Writing $\mathcal G_p$9 if $p = 2$0 and $p = 2$1 otherwise, and defining auxiliary quantities $p = 2$2, $p = 2$3 (and primed analogues), the paper establishes four necessary conditions:

• $p = 2$4,
• $p = 2$5,
• each of $p = 2$6 and $p = 2$7 divides four explicit integer combinations of the vector entries,
• the pair satisfies one of the inequalities $p = 2$8 or $p = 2$9.

These follow from the rigid form of matrices in $\beta$0: if $\beta$1 then the auxiliary ratios $\beta$2 must equal specific entries of $\beta$3 (e.g., $\beta$4, $\beta$5), forcing $\beta$6.

The main classification theorem then enumerates seven explicit normal forms that $\beta$7 and its image must satisfy — including trivial stabilization-type moves, transvections in $\beta$8 or $\beta$9 directions, and more complicated families parameterized by integers $\rho$0 or $\rho$1 subject to the determinant condition $\rho$2. Two cases arise only at $\rho$3 or require the $\rho$4 inequalities. The proof proceeds by setting up eigenvalue equations for the matrix $\rho$5 over $\rho$6 and performing an exhaustive row-reduction case analysis; notably, since $\rho$7, invertibility fails over $\rho$8, and integral solutions exist only in special circumstances.

As an application, the authors note that knots lying in the genus one surface except for one arc — yielding vectors of the form $\rho$9 — admit only a short list of possible homological images under the Goeritz action, giving a concrete sieve for identifying topologically equivalent positions relevant to lens space surgery questions.

Homotopy obstruction

The kernel theorem also yields a clean topological obstruction: if $\gamma$0 and $\gamma$1 are homologous but not freely homotopic in $\gamma$2, or not freely homotopic in $\gamma$3, they cannot be Goeritz equivalent. The proof shows that two homologous curves are Goeritz equivalent precisely when related by reducing sphere twists (conjugates of $\gamma$4), using Cho–Koda's correspondence between dual disk pairs and reducing spheres together with transitivity of the Goeritz action on primitive disks. Reducing sphere twists evidently preserve free homotopy classes in both handlebodies. This mirrors the analogous result for $\gamma$5 from the authors' earlier paper, though the verification requires the lens-space-specific machinery developed above.

Limitations and open questions

The results are necessarily restricted to $\gamma$6, where Cho's presentations are available; extending to general lens spaces $\gamma$7 would require new group presentations, and the paper leaves this untouched. The obstructions provided are necessary but not sufficient: the paper does not prove converses establishing when satisfying conditions guarantee Goeritz equivalence, so the framework identifies candidates rather than certifying equivalence. The enumeration in the classification theorem assumes $\gamma$8; the degenerate case $\gamma$9 (where $L(p,1)$00 is non-invertible even over $L(p,1)$01) is excluded from the analysis. Finally, whether the inequality distinguishing $L(p,1)$02 reflects deeper geometric phenomena about the Goeritz groups, or is an artifact of the presentation, is not addressed.

Conclusion

This paper transfers the homological obstruction program for genus two Goeritz equivalence from $L(p,1)$03 to $L(p,1)$04, computing the full star representation, determining its kernel (the reducing sphere twists $L(p,1)$05) and its image exactly as a function of $L(p,1)$06, and deriving both arithmetic conditions on homology vectors and a fundamental-group-theoretic obstruction for curve equivalence. The results provide a practical computational filter for distinguishing knot positions on the genus two Heegaard surface of $L(p,1)$07, with direct relevance to studying lens space surgeries and the surrounding variants of the Berge conjecture.