A calculus for branched spine of 3-manifolds

Branched Spines and Contact Structures on 3-manifolds

We introduce and analyze the characteristic foliation induced by a contact structure on a branched surface, in particular a branched standard spine of a 3-manifold. We extend to (fairly general) singular foliations of branched surfaces the local existence and uniqueness results which hold for genuine surfaces. Moreover we show that global uniqueness holds when restricting to tight structures. We establish branched versions of the elimination lemma. We prove a smooth version of the Gillman-Rolfsen PL-embedding theorem, deducing that branched spines can be used to construct contact structures in…

Branched shadows and complex structures of 4-manifolds

We define and study branched shadows of 4-manifolds as a combination of branched spines of 3-manifolds and Turaev's shadows. We use these objects to combinatorially represent 4-manifolds equipped with $Spin^c$-structures and homotopy classes of almost complex structures. We then use branched shadows to study complex 4-manifolds and prove that each almost complex structure on a 4-handlebody is homotopic to a complex one.

关于分支配边的 Matveev-Piergallini 移动

On Matveev-Piergallini moves for branched spines

本文证明了分支配边上的16种 Matveev-Piergallini 移动可由基本 MP 移动和纯滑动移动导出，从而简化了闭 $3$-流形和梳状 $3$-流形的组合描述，并扩展至带框和旋量 $3$-流形，有助于理解量子不变量。

S-stable foliations on flow-spines with transverse Reeb flow

The notion of S-stability of foliations on branched simple polyhedrons is introduced by R. Benedetti and C. Petronio in the study of characteristic foliations of contact structures on 3-manifolds. We additionally assume that the 1-form $β$ defining a foliation on a branched simple polyhedron $P$ satisfies $dβ>0$, which means that the foliation is a characteristic foliation of a contact form whose Reeb flow is transverse to $P$. In this paper, we show that if there exists a 1-form $β$ on $P$ with $dβ>0$ then we can find a 1-form with the same property and additionally being S-stable. We then pr…

Construction of spines of two-bridge link complements and upper bounds of their Matveev complexities

We give upper bounds of the Matveev complexities of two-bridge link complements by constructing their spines explicitly. In particular, we determine the complexities for an infinite sequence of two-bridge links corresponding to the continued fractions of the form [2,1,...,1,2]. We also give upper bounds for the 3-manifolds obtained as meridian-cyclic branched coverings of the 3-sphere along two-bridge links.

Covering moves and Kirby calculus

We show that simple coverings of B^4 branched over ribbon surfaces up to certain local ribbon moves bijectively represent orientable 4-dimensional 2-handlebodies up to handle sliding and addition/deletion of cancelling handles. As a consequence, we obtain an equivalence theorem for simple coverings of S^3 branched over links, in terms of local moves. This result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of our equivalence theorem to possibly non-simple coverings of S^3 br…