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Daiki Iguchi

Publications and source records attributed to Daiki Iguchi.

6 recordsLinked to original sources
arXiv2026

A proof of Powell's conjecture on the Goeritz group of $S^3$

For a genus $g$ Heegaard splitting of the $3$-sphere, the Goeritz group is defined to be the group of isotopy classes of diffeomorphisms of the $3$-sphere that preserve the splitting setwise. In this paper, we prove the following conjecture proposed by Powell: For every $g \ge 3$, the Goeritz group of a genus $g$ Heegaard splitting is generated by four specific elements. Our proof relies crucially on the fact that a Heegaard surface of the $3$-sphere is topologically minimal, that is, its disk complex has nontrivial homotopy group in some dimension. Along the way, we also give a new proof of the fact that a genus $g$ Heegaard surface of the $3$-sphere has topological index $2g-1$.

math.GT↗
arXiv2025

Thick isotopy property and the mapping class groups of Heegaard splittings

We give a necessary and sufficient condition for the fundamental group of the space of Heegaard splittings of an irreducible $3$-manifold to be finitely generated. The condition is exactly the conclusion of the thick isotopy lemma proved by Colding, Gabai and Ketover, which says that any isotopy of a Heegaard surface is achieved by a $1$-parameter family of surfaces with area bounded above by a universal constant and with some ``thickness property''. We also prove that a Heegaard splitting of a hyperbolic or spherical $3$-manifold satisfies the condition if it is topologically minimal (in the sense of Bachman) and its disk complex has finitely generated homotopy group. In conclusion, such a Heegaard splitting has finitely generated mapping class group.

math.GT↗
arXiv2020

Goeritz groups of bridge decompositions

For a bridge decomposition of a link in the $3$-sphere, we define the Goeritz group to be the group of isotopy classes of orientation-preserving homeomorphisms of the $3$-sphere that preserve each of the bridge sphere and link setwise. After describing basic properties of this group, we discuss the asymptotic behavior of the minimal pseudo-Anosov entropies. This gives an application to the asymptotic behavior of the minimal entropies for the original Goeritz groups of Heegaard splittings of the $3$-sphere and the real projective space.

math.GT↗
arXiv2020

Twisted book decompositions and the Goeritz groups

We consider the Goeritz groups of the Heegaard splittings induced from twisted book decompositions. We show that there exist Heegaard splittings of distance $2$ that have the infinite-order mapping class groups whereas that are not induced from open book decompositions. Explicit computation of those mapping class groups are given.

math.GT↗