Surface bundles versus Heegaard splittings

Non-isotopic Heegaard splittings of Seifert fibered spaces

We find a geometric invariant of isotopy classes of strongly irreducible Heegaard splittings of toroidal 3-manifolds. Combining this invariant with a theorem of R Weidmann, proved here in the appendix, we show that a closed, totally orientable Seifert fibered space M has infinitely many isotopy classes of Heegaard splittings of the same genus if and only if M has an irreducible, horizontal Heegaard splitting, has a base orbifold of positive genus, and is not a circle bundle. This characterizes precisely which Seifert fibered spaces satisfy the converse of Waldhausen's conjecture.

Heegaard Floer homology and genus one, one boundary component open books

We compute the Heegaard Floer homology of any rational homology 3-sphere with an open book decomposition of the form (T,ϕ), where T is a genus one surface with one boundary component. In addition, we compute the Heegaard Floer homology of any T^2-bundle over S^1 with first Betti number equal to one, and we compare our results with those of Lebow on the embedded contact homology of such torus bundles. We use these computations to place restrictions on Stein-filllings of the contact structures compatible with such open books, to narrow down somewhat the class of 3-braid knots with finite concord…

Turaev-Viro TQFT and the Rank versus Genus Conjecture

This paper presents a way to estimate the Heegaard genus of a $3$-manifold using the Turaev-Viro state sum TQFT. The Turaev-Viro state sum TQFT is derived from the modular category associated to the quantum group $U_q(\mathfrak{sl}_2)$, which is unitary for some $q$ by Wenzl. Hence by Turaev and Virelizier the corresponding TQFT is unitary. We modify a proof by Garoufalidis to give a lower bound of the Heegaard genus using a unitary TQFT, and then use the software Regina to provide some known counterexamples to the rank versus genus conjecture.

On the Heegaard Floer homology of a surface times a circle

We make a detailed study of the Heegaard Floer homology of the product of a closed surface Sigma_g of genus g with S^1. We determine HF^+ for this 3-manifold completely for the spin^c structure having trivial first Chern class, which for g>2 was previously unknown. We show that in this case HF^\infty is closely related to the cohomology of the total space of a certain circle bundle over the Jacobian torus of Sigma_g, and furthermore that HF^+ of a surface times a circle with integral coefficients contains nontrivial 2-torsion whenever g>2. This is the first example known to the authors of tors…

Detecting surface bundles in finite covers of hyperbolic closed 3-manifolds

The main theorem of this article provides sufficient conditions for a degree $d$ finite cover $M'$ of a hyperbolic 3-manifold $M$ to be a surface-bundle. Let $F$ be an embedded, closed and orientable surface of genus $g$, close to a minimal surface in the cover $M'$, splitting $M'$ into a disjoint union of $q$ handlebodies and compression bodies. We show that there exists a fiber in the complement of $F$ provided that $d$, $q$ and $g$ satisfy some inequality involving an explicit constant $k$ depending only on the volume and the injectivity radius of $M$. In particular, this theorem applies to…

Constructions of surface bundles with rank two fundamental groups

We give a construction of hyperbolic 3-manifolds with rank two fundamental groups and report an experimental search to find such manifolds. Our manifolds are all surface bundles over the circle with genus two surface fiber. For the manifolds so obtained, we then examine whether they are of Heegaard genus two or not. As a byproduct, we give an infinite family of fibered knots of genus two in the 3-sphere whose knot groups are of rank two.

Homotopy motions of surfaces in $3$-manifolds

We introduce the concept of a homotopy motion of a subset in a manifold, and give a systematic study of homotopy motions of surfaces in closed orientable 3-manifolds. This notion arises from various natural problems in 3-manifold theory such as domination of manifold pairs, homotopical behavior of simple loops on a Heegaard surface, and monodromies of virtual branched covering surface bundles associated to a Heegaard splitting.

曲面丛和圆上丛的多剖分

Multisections of surface bundles and bundles over the circle

本文证明曲面丛和纤维已多剖分的圆上纤维丛在任意维度均承认多剖分，并给出显式构造与示例。

On hyperbolic surface bundles over the circle as branched double covers of the $3$-sphere

The branched virtual fibering theorem by Sakuma states that every closed orientable $3$-manifold with a Heegaard surface of genus $g$ has a branched double cover which is a genus $g$ surface bundle over the circle. It is proved by Brooks that such a surface bundle can be chosen to be hyperbolic. We prove that the minimal entropy over all hyperbolic, genus $g$ surface bundles as branched double covers of the $3$-sphere behaves like 1/$g$. We also give an alternative construction of surface bundles over the circle in Sakuma's theorem when closed $3$-manifolds are branched double covers of the $3…

Rank and genus of 3-manifolds

We construct a counterexample to the Rank versus Genus Conjecture, i.e. a closed orientable hyperbolic 3-manifold with rank of its fundamental group smaller than its Heegaard genus. Moreover, we show that the discrepancy between rank and Heegaard genus can be arbitrarily large for hyperbolic 3-manifolds. We also construct toroidal such examples containing hyperbolic JSJ pieces.

Rational cuspidal curves in projective surfaces. Topological versus algebraic obstructions

We study rational cuspidal curves in projective surfaces. We specify two criteria obstructing possible configurations of singular points that may occur on such curves. One criterion generalizes the result of Fernandez de Bobadilla, Luengo, Melle--Hernandez and Nemethi and is based on the Bezout theorem. The other one is a generalization of the result obtained by Livingston and the author and relies on Ozsvath--Szabo inequalities for $d$-invariants in Heegaard Floer homology. We show by means of explicit calculations that the two approaches give very similar obstructions.

虚拟同调挠：$I$-丛书中的丰富性与增长

Virtual homological torsion: abundance versus growth in books of $I$-bundles

本文研究 $I$-丛书的虚拟同调挠：证明正则覆叠塔中挠次指数增长，而非正则覆叠塔中挠可指数增长，揭示覆叠正则性对挠增长的影响。

Symplectic fillability of tight contact structures on torus bundles

We study weak versus strong symplectic fillability of some tight contact structures on torus bundles over the circle. In particular, we prove that almost all of these tight contact structures are weakly, but not strongly symplectically fillable. For the 3-torus this theorem was established by Eliashberg.

双仿射 Hecke 代子多项式表示的 Floer 理论解释

A Floer-theoretic interpretation of the polynomial representation of the double affine Hecke algebra

本文构造了曲面余切丛中余切纤维与曲线余法丛的缠绕高维 Heegaard Floer 同调与辫子 skein 群间的同构，证明其构成曲面 Hecke 代数上的右模，并在环面情形下等价于 DAHA 的多项式表示。