Homology cylinders and sutured manifolds for homologically fibered knots

关于 sutured 流形中的 Heegaard Floer 极小纽结

On Heegaard Floer minimal knots in sutured manifolds

本文在 Heegaard Floer 同调框架下分类了平衡 sutured 流形中的极小纽结，与 instanton Floer 分类一致，支持两种同调同构的猜想，并研究了 $S^1\ imes S^2$ 中的链接 Floer 同调植物学问题。

Sutured Floer homology, fibrations, and taut depth one foliations

For an oriented irreducible 3-manifold M with non-empty toroidal boundary, we describe how sutured Floer homology ($SFH$) can be used to determine all fibered classes in $H^1(M)$. Furthermore, we show that the $SFH$ of a balanced sutured manifold $(M,γ)$ detects which classes in $H^1(M)$ admit a taut depth one foliation such that the only compact leaves are the components of $R(γ)$. The latter had been proved earlier by the first author under the extra assumption that $H_2(M)=0$. The main technical result is that we can obtain an extremal $\text{Spin}^c$-structure $\mathfrak{s}$ (i.e., one tha…

通用 $L^2$- torsion 和 3-流形的缝合分解

Universal $L^2$-torsion and sutured decomposition for 3-manifolds

本文研究了 3-流形的通用 $L^2$-torsion 及其在缝合分解中的应用，证明了该不变量能检测纤维性并推导了分解公式。

Abelian quotients of monoids of homology cylinders

A homology cylinder over a surface consists of a homology cobordism between two copies of the surface and markings of its boundary. The set of isomorphism classes of homology cylinders over a fixed surface has a natural monoid structure and it is known that this monoid can be seen as an enlargement of the mapping class group of the surface. We now focus on abelian quotients of this monoid. We show that both the monoid of all homology cylinders and that of irreducible homology cylinders are not finitely generated and moreover they have big abelian quotients. These properties contrast with the f…

亏格为一的近纤维链环

Nearly fibered links with genus one

本文分类了三维球面中满足特定欧拉示性数条件的近纤维 n 分量链环，其最大 Alexander 分次秩为二，证明受 Baldwin 和 Sivek 启发，并利用缝合 Floer 同调技术，同时计算了这些链环的链环 Floer 同调群。

瞬子2- torsion与纤维化结

Instanton 2-torsion and fibered knots

研究证明了纤维化结的瞬子2-torsion性质，并通过缝合瞬子理论得到相关同调的公式，同时揭示了Alexander多项式与瞬子同调的关系及对Heegaard Floer理论的联系。

扭转的亚历山大多项式、绳索同调柄结构以及图松规范

Twisted Alexander Polynomials, Ribbon Homology Cobordisms, and the Thurston Norm

本文利用扭转亚历山大多项式证明，若存在从Y_-到Y_+的绳索同调柄结构且Y_-不可约，则Y_-的图松规范单位球包含Y_+的。同时证明Y_+的纤维类对应于Y_-的。

Non ambiguous structures on 3-manifolds and quantum symmetry defects

The state sums defining the quantum hyperbolic invariants (QHI) of hyperbolic oriented cusped $3$-manifolds can be split in a "symmetrization" factor and a "reduced" state sum. We show that these factors are invariants on their own, that we call "symmetry defects" and "reduced QHI", provided the manifolds are endowed with an additional "non ambiguous structure", a new type of combinatorial structure that we introduce in this paper. A suitably normalized version of the symmetry defects applies to compact $3$-manifolds endowed with $PSL_2(\mathbb{C})$-characters, beyond the case of cusped manifo…