Automorphisms of the 3-sphere that preserve spatial graphs and handlebody-knots

We consider the group of isotopy classes of automorphisms of the 3-sphere that preserve a spatial graph or a handlebody-knot embedded in it. We prove that the group is finitely presented for an arbitrary spatial graph or a reducible handlebody-knot of genus two. We also prove that the groups for "most" irreducible genus two handlebody-knots are finite.Comment: 23 pages, 11 figures; minor changes, typos corrected; to appear in Mathematical Proceedings of the Cambridge Philosophical Societ

Stable maps and branched shadows of 3-manifolds

Turaev's shadow can be seen locally as the Stein factorization of a stable map. In this paper, we define the notion of stable map complexity for a compact orientable 3-manifold bounded by (possibly empty) tori counting, with some weights, the minimal number of singular fibers of codimension 2 of stable maps into the real plane, and prove that this number equals the minimal number of vertices of its branched shadows. In consequence, we give a complete characterization of hyperbolic links in the 3-sphere whose exteriors have stable map complexity 1 in terms of Dehn surgeries, and also give an observation concerning the coincidence of the stable map complexity and shadow complexity using estimations of hyperbolic volumes.Comment: 45 pages, 45 figure

Shadows of acyclic 4-manifolds with sphere boundary

In terms of Turaev's shadows, we provide a sufficient condition for a compact, smooth, acyclic 4-manifold with boundary the 3-sphere to be diffeomorphic to the standard 4-ball. As a consequence, we prove that if a compact, smooth, acyclic 4-manifold with boundary the 3-sphere has shadow-complexity at most 2, then it is diffeomorphic to the standard 4-ball.Comment: 14 pages, 12 figures, plus appendice

Disk complexes and genus two Heegaard splittings for non-prime 3-manifolds

Given a genus two Heegaard splitting for a non-prime 3-manifold, we define a special subcomplex of the disk complex for one of the handlebodies of the splitting, and then show that it is contractible. As applications, first we show that the complex of Haken spheres for the splitting is contractible, which refines the results of Lei and Lei-Zhang. Secondly, we classify all the genus two Heegaard splittings for non-prime 3-manifolds, which is a generalization of the result of Montesinos-Safont. Finally, we show that the mapping class group of the splitting, called the Goeritz group, is finitely presented by giving its explicit presentation.Comment: 22 pages, 11 figures; proofs improved, typos corrected; to appear in International Mathematics Research Notice

Haken spheres for genus two Heegaard splittings

A manifold which admits a reducible genus-$2$ Heegaard splitting is one of the $3$-sphere, $S^2 \times S^1$, lens spaces or their connected sums. For each of those splittings, the complex of Haken spheres is defined. When the manifold is the $3$-sphere, $S^2 \times S^1$ or the connected sum whose summands are lens spaces or $S^2 \times S^1$, the combinatorial structure of the complex has been studied by several authors. In particular, it was shown that those complexes are all contractible. In this work, we study the remaining cases, that is, when the manifolds are lens spaces. We give a precise description of each of the complexes for the genus-$2$ Heegaard splittings of lens spaces. A remarkable fact is that the complexes for most lens spaces are not contractible and even not connected.Comment: 10 pages, 4 figure

The disk complex and 2-bridge knots

We give an alternative proof of a result of Kobayashi and Saeki that every genus one $1$-bridge position of a non-trivial $2$-bridge knot is a stabilization.Comment: 10 pages, 4 figure

Primitive disk complexes for lens spaces

For a genus two Heegaard splitting of a lens space, the primitive disk complex is defined to be the full subcomplex of the disk complex for one of the handlebodies of the splitting spanned by all vertices of primitive disks. In this work, we describe the complete combinatorial structure of the primitive disk complex for the genus two Heegaard splitting of each lens space. In particular, we find all lens spaces whose primitive disk complexes are contractible.Comment: 26 pages, 11 figure

Extending automorphisms of the genus-2 surface over the 3-sphere

An automorphism $f$ of a closed orientable surface $\Sigma$ is said to be extendable over the 3-sphere $S^3$ if $f$ extends to an automorphism of the pair $(S^3, \Sigma)$ with respect to some embedding $\Sigma \hookrightarrow S^3$. We prove that if an automorphism of a genus-2 surface $\Sigma$ is extendable over $S^3$, then $f$ extends to an automorphism of the pair $(S^3, \Sigma)$ with respect to an embedding $\Sigma \hookrightarrow S^3$ such that $\Sigma$ bounds genus-2 handlebodies on both sides. The classification of essential annuli in the exterior of genus-2 handlebodies embedded in $S^3$ due to Ozawa and the second author plays a key role.Comment: 23 pages, 16 figures; typos and errors correcte

The genus two Goeritz group of S2×S1S^2 \times S^1

The genus-g Goeritz group is the group of isotopy classes of orientation-preserving homeomorphisms of a closed orientable 3-manifold that preserve a given genus-g Heegaard splitting of the manifold. In this work, we show that the genus-2 Goeritz group of $S^2 \times S^1$ is finitely presented, and give its explicit presentation.Comment: minor changes; to appear in Mathematical Research Letter

Connected primitive disk complexes and genus two Goeritz groups of lens spaces

Given a stabilized Heegaard splitting of a $3$-manifold, the primitive disk complex for the splitting is the subcomplex of the disk complex for a handlebody in the splitting spanned by the vertices of the primitive disks. In this work, we study the structure of the primitive disk complex for the genus two Heegaard splitting of each lens space. In particular, we show that the complex for the genus two splitting for the lens space $L(p, q)$ with $1\leq q \leq p/2$ is connected if and only if $p \equiv \pm 1 \pmod q$, and describe the combinatorial structure of each of those complexes. As an application, we obtain a finite presentation of the genus two Goeritz group of each of those lens spaces, the group of isotopy classes of orientation preserving homeomorphisms of the lens space that preserve the genus two Heegaard splitting of it.Comment: 32 pages, 17 figures; This is an extended version of our earlier preprint arXiv:1206.6243 "Primitive disk complexes for lens spaces", which deals with the structure of the primitive disk complexes for lens spaces. The arguments are polished under a new organization, and further a new application is combine