We investigate the complexity of finding an embedded non-orientable surface of Euler genus $g$ in a triangulated $3$-manifold. This problem occurs both as a natural question in low-dimensional topology, and as a first non-trivial instance of embeddability of complexes into $3$-manifolds.
We prove that the problem is NP-hard, thus adding to the relatively few hardness results that are currently known in 3-manifold topology. In addition, we show that the problem lies in NP when the Euler genus g is odd, and we give an explicit algorithm in this case.
We study trisections of smooth, compact non-orientable 4-manifolds, and introduce trisections of non-orientable 4-manifolds with boundary. In particular, we prove a non-orientable analogue of a classical theorem of Laudenbach-Poénaru. As a consequence, trisection diagrams and Kirby diagrams of closed non-orientable 4-manifolds exist. We discuss how the theory of trisections may be adapted to the setting of non-orientable 4-manifolds with many examples.
The construction of integer linking numbers of closed curves in a three-dimensional manifold usually appeals to the orientation of this manifold. We discuss how to avoid it constructing similar homotopy invariants of links in non-orientable manifolds.
A closed orientable 3-manifold with distinct distance three genus two Heegaard splittings
本文给出了一个闭可定向三维流形的例子，它具有不同距离的三亏格两Heegaard分裂.这证明了Rubinstein和Scharlemann在1998年的论文Genus Two Heegaard Splittings of Orientable 3-Manifolds中描述的闭可定向3-流形的交替亏格2 Heegaard分裂的构造，并不产生所有的交替亏格2分裂，必须增广。
We give an upper bound for the Matveev complexity of the whole class of closed connected orientable prime graph manifolds that is sharp for all 14502 graph manifolds of the Recognizer catalogue (available at \texttt{http://matlas.math.csu.ru/?page=search}).
This paper extends the results from the author's previous paper to consider finite, fiber- and orientation- preserving group actions on closed, orientable Seifert manifolds $M$ that fiber over a non-orientable base space. An orientable base space double cover $\tilde{M}$ of $M$ is constructed and then an isomorphism between the fiber- and orientation-preserving diffeomorphisms of ${M}$ and the fiber- and orientation-preserving actions on $\tilde{M}$ that preserve the orientation on the fibers and commute with the covering translation is shown. This result and previous results lead to a constru…
In this paper we consider the finite groups that act fiber- and orientation-preservingly on closed, compact, and orientable Seifert manifolds that fiber over an orientable base space. We establish a method of constructing such group actions and then show that if an action satisfies a condition on the obstruction class of the Seifert manifold, it can be derived from the given construction. The obstruction condition is refined and the general structure of the finite groups that act via the construction is provided.
We find conditions under which a non-orientable closed surface S embedded into an orientable closed 4-manifold X can be represented by a connected sum of an embedded closed surface in X and an unknotted projective plane in a 4-sphere. This allows us to extend the Gabai 4-dimensional light bulb theorem and the Auckly-Kim-Melvin-Ruberman-Schwartz "one is enough" theorem to the case of non-orientable surfaces.
We describe a general method to compute the $\mathbb{Z}_2$-Thurston norm for every $\mathbb{Z}_2$-homology class in an orientable Seifert manifold with orientable orbit surface. Our main tools are pseudo-horizontal surfaces. We give a necessary and sufficient criterion for the existence of pseudo-horizontal surfaces, calculate the non-orientable genera for such surfaces, and detect their $\mathbb{Z}_2$-homology classes. We then describe an algorithm to calculate the $\mathbb{Z}_2$-Thurston norm of each $\mathbb{Z}_2$-homology classes. We also present several interesting examples.
In this paper, it is shown that every orientable closed 3-manifold maps with nonzero degree onto at most finitely many homeomorphically distinct irreducible non-geometric orientable closed 3-manifolds. Moreover, given any nonzero integer, as a mapping degree up to sign, every orientable closed 3-manifold maps with that degree onto only finitely many homeomorphically distinct orientable closed 3-manifolds.
Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. This paper improves and implements an algorithm due to Bell, Hass, Rubinstein and Tillmann to compute trisections using triangulations, and extends it to non-orientable 4-manifolds. Lower bounds on trisection genus are given in terms of Betti numbers and used to determine the trisection genus of all standard simply connected PL 4-manifolds. In addition, we construct trisections of small genus directly from the sim…
Thurston's hyperbolization theorem for Haken manifolds and normal surface theory yield an algorithm to determine whether or not a compact orientable 3-manifold with nonempty boundary consisting of tori admits a complete finite-volume hyperbolic metric on its interior. A conjecture of Gabai, Meyerhoff, and Milley reduces to a computation using this algorithm.
我们证明了每个可定向的3-流形都是一个分类空间BΓ，其中Γ是R的同胚芽的广群。这表明每个可定向的3-流形M允许余维为1的叶化F，使得每个叶的完整覆盖是可收缩的。我们构造的F可以取为C^1，但不能取为C^2。这样一个F的存在肯定地回答了Tsuboi [Classifying spaces for groupoid structures，notes from minicourse at PUC，Rio de Janeiro（2001）]提出的一个问题，但是对于某些C^ \infty groupoid Γ，M是否= BΓ的问题没有解决。
We prove that for any compact orientable connected 3-manifold with torus boundary, a concatenation of it and the direct product of the circle and the Klein bottle with an open 2-disk removed admits a Lagrangian embedding into the standard symplectic 6-space. Moreover, minimal Maslov number of the Lagrangian embedding is equal to 1.
Since there is no hyperbolic Dehn filling theorem for higher dimensions, it is challenging to construct explicit hyperbolic manifolds of small volume in dimension at least four. Here, we build up closed hyperbolic 4-manifolds of volume $\frac{34π^2}{3}\cdot 16$ by using the small cover theory. In particular, we classify all of the orientable four-dimensional small covers over the right-angled 120-cell up to homeomorphism; these are all with even intersection forms.
The Triple Point Spectrum of closed orientable 3-manifolds
定义了闭3-流形的三重点数和三重点谱。Vigara，Representación de 3-variedades por esferas de Dehn rellenplant，PhD Thesis，UNED 2006）.它们是拓扑不变量，使用最小填充Dehn曲面的三重点的数量来度量三维流形的复杂性。给出了这些不变量的基本性质，并计算了$\mathbb{S}^2\times \mathbb{S}^1$和$\mathbb{S}^3$的三重点谱.
It is well-known that odd-dimensional manifolds have Euler characteristic zero. Furthemore orientable manifolds have an even Euler characteristic unless the dimension is a multiple of $4$. We prove here a generalisation of these statements: a $k$-orientable manifold (or more generally Poincaré complex) has even Euler characteristic unless the dimension is a multiple of $2^{k+1}$, where we call a manifold $k$-orientable if the $i^{th}$ Stiefel-Whitney class vanishes for all $0<i< 2^k$ ($k\geq 0$). More generally, we show that for a $k$-orientable manifold the Wu classes $v_l$ vanish for all $l$…
It is proved that the Hasse-Weil zeta functions of the canonical components of the ${\rm SL}_2$ (${\rm PSL}_2$)-character varieties of closed orientable complete hyperbolic $3$-manifolds of finite volume are equal to the Dedekind zeta functions of their trace fields (invariant trace fields). When the closed $3$-manifold is arithmetic, the special value at $s=2$ of the Hasse-Weil zeta function of the canonical component of the ${\rm PSL}_2$-character variety is expressed in terms of the hyperbolic volume of the manifold up to rational numbers.
In this paper we study existence and lack thereof of closed embedded orientable co-dimension one totally geodesic submanifolds of minimal volume cusped orientable hyperbolic manifolds.
We show that for any nonzero class $A$ in $H_2(X; \mathbb{Z}_2)$ in a rational 4-manifold $X$, $A$ is represented by a nonorientable embedded Lagrangian surface L (for some symplectic structure) if and only if $P(A)\equiv (L) (mod\ 4)$; where $P(A)$ denotes the mod 4 valued Pontrjagin square of $A$.
In this paper we deal with Seifert fibre spaces, which are compact 3-manifolds admitting a foliation by circles. We give a combinatorial description for these manifolds in all the possible cases: orientable, non-orientable, closed, with boundary. Moreover, we compute a potentially sharp upper bound for their complexity in terms of the invariants of the combinatorial description, extending to the non-orientable case results by Fominykh and Wiest for the orientable case with boundary and by Martelli and Petronio for the closed orientable case.
There are only 10 Euclidean forms, that is flat closed three dimensional manifolds: six are orientable $\mathcal{G}_1,\dots,\mathcal{G}_6$ and four are non-orientable $\mathcal{B}_1,\dots,\mathcal{B}_4$. The aim of this paper is to describe all types of $n$-fold coverings over the non-orientable Euclidean manifolds $\mathcal{B}_{3}$ and $\mathcal{B}_{4}$, and calculate the numbers of non-equivalent coverings of each type. The manifolds $\mathcal{B}_{3}$ and $\mathcal{B}_{4}$ are uniquely determined among non-orientable forms by their homology groups $H_1(\mathcal{B}_{3})=\ZZ_2\times \ZZ_2 \tim…
Considering a solid 3-dimensional Klein bottle and a collaring of its boundary, can we extend a generic $C^\infty$ non-singular function defined on the collaring to the full solid Klein bottle without critical points? We give a condition on the Reeb graph of the given function that is necessary and sufficient for the existence of such a non-singular extension.
