Embeddings of $4$--manifolds in $\CP^3$

具有可定向基底的Seifert 3-流形的Dijkgraaf-Witten不变量

The Dijkgraaf-Witten invariants of Seifert 3-manifolds with orientable bases

本文针对具有可定向基底的可定向Seifert 3-流形，推导出其Dijkgraaf-Witten不变量的计算公式，为相关拓扑不变量研究提供了显式结果。

The complexity of orientable graph manifolds

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}).

Heegaard splittings of Seifert manifolds without an orientable base space

We use Heath's, Moriah's and Schultens's results to prove that irreducible Heegaard splittings of orientable Seifert manifolds with nonorientable base space are either vertical or horizontal.

Orientable closed 3-manifolds with surface-complexity one

After a short summary of known results on surface-complexity of closed 3-manifolds, we will classify all closed orientable 3-manifolds with surface-complexity one.

曲面复杂度为一的不可定向3-流形

Non-orientable 3-manifolds of surface-complexity one

本文分类了通过粘合立方体面得到的闭不可定向P²-不可约3-流形，证明它们恰为四个平坦流形，即曲面复杂度为一的情形。

Finite, fiber- and orientation-preserving actions on orientable Seifert manifolds with non-orientable base space

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…

Surgery Description of Orientation-Preserving $D_{2p}$-Actions on Compact Orientable 3-Manifolds

This paper has been withdrawn because its contents have become subsumed in Section 5.2 of arXiv:0805.2307.

可定向紧边双曲3流形的完整10四面体普查

The complete $10$-tetrahedra census of orientable cusped hyperbolic $3$-manifolds

本研究通过普查10个四面体的双曲3流形，发现新的流形及例外Dehn填充，并揭示了新的最简单双曲结外皮。

Non-orientable 3-manifolds of small complexity

We classify all closed non-orientable P2-irreducible 3-manifolds having complexity up to 6 and we describe some having complexity 7. We show in particular that there is no such manifold with complexity less than 6, and that those having complexity 6 are precisely the 4 flat non-orientable ones and the filling of the Gieseking manifold, which is of type Sol. The manifolds having complexity 7 we describe are Seifert manifolds of type H2 x S1 and a manifold of type Sol.

Non-orientable 3-manifolds of complexity up to 7

We classify all closed non-orientable P2-irreducible 3-manifolds with complexity up to 7, fixing two mistakes in our previous complexity-up-to-6 classification. We show that there is no such manifold with complexity less than 6, five with complexity 6 (the four flat ones and the filling of the Gieseking manifold, which is of type Sol, and three with complexity 7 (one manifold of type Sol, and the two manifolds of type H2xR with smallest base orbifolds).

Finite, fiber- and orientation-preserving group actions on totally orientable Seifert manifolds

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.

On non-orientable surfaces in 4-manifolds

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.

A catalogue of orientable 3-manifolds triangulated by 30 coloured tetrahedra

The present paper follows the computational approach to 3-manifold classification via edge-coloured graphs, already performed by several authors with respect to orientable 3-manifolds up to 28 coloured tetrahedra, non-orientable 3-manifolds up to 26 coloured tetrahedra, genus two 3-manifolds up to 34 coloured tetrahedra: in fact, by automatic generation and analysis of suitable edge-coloured graphs, called crystallizations, we obtain a catalogue of all orientable 3-manifolds admitting coloured triangulations with 30 tetrahedra. These manifolds are unambiguously identified via JSJ decomposition…

Guts 与具有 3 个尖点的最小体积可定向双曲 3-流形

Guts and The Minimal Volume Orientable Hyperbolic 3-Manifold with 3 Cusps

本文借助缝合流形与 pared 流形中的 guts，证明满足所有二阶同调类均为 libroid 的三尖点可定向双曲 3-流形体积不低于 6 倍 Catalan 常数。

On $\mathbb{Z}_2$-Thurston norms and pseudo-horizontal surfaces in orientable Seifert $3$-manifolds

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.

关于可定向3-流形可平行化的一个简短证明的注记

A note on a short proof of the parallelizability of orientable $3$-manifolds

本文综述并补全了利用纽结理论证明Stiefel定理（所有可定向3-流形可平行化）的简短证明，借助切丛与法丛关系完成证明，并给出7-流形的相关注记与反例。

Finiteness of nonzero degree maps between three-manifolds

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.

The minimal volume orientable hyperbolic 3-manifold with 4 cusps

We prove that the 8^4_2 link complement is the minimal volume orientable hyperbolic manifold with 4 cusps. Its volume is twice of the volume V_8 of the ideal regular octahedron, i.e. 7.32... = 2V_8. The proof relies on Agol's argument used to determine the minimal volume hyperbolic 3-manifolds with 2 cusps. We also need to estimate the volume of a hyperbolic 3-manifold with totally geodesic boundary which contains an essential surface with non-separating boundary.

On the subadditivity of Montesinos complexity of closed orientable 3-manifolds

A filling Dehn sphere $Σ$ in a closed 3-manifold $M$ is a sphere transversely immersed in $M$ that defines a cell decomposition of $M$. Every closed 3-manifold has a filling Dehn sphere. The Montesinos complexity of a $3$-manifold $M$ is defined as the minimal number of triple points among all the filling Dehn spheres of $M$. A sharp upper bound for the Montesinos complexity of the connected sum of two 3-manifolds is given.

The linking pairings of orientable Seifert manifolds

We compute the $p$-primary components of the linking pairings of orientable 3-manifolds admitting a fixed-point free $S^1$-action. Using this, we show that any non-singular linking pairing on a finite abelian group with homogeneous 2-primary summand is realized by such a manifold. However, some pairings on inhomogeneous 2-groups are not realizable by any Seifert fibred 3-manifold.

The minimal volume orientable hyperbolic 2-cusped 3-manifolds

We prove that the Whitehead link complement and the (-2, 3, 8) pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume 3.66... = 4 x Catalan's constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on volume and strong constraints on the manifolds that realize that lower bound.

Non-orientable genus of knots in punctured Spin 4-manifolds

For a closed 4-manifold $X$ and a knot $K$ in the boundary of punctured $X$, we define $γ_X^0(K)$ to be the smallest first Betti number of non-orientable and null-homologous surfaces in punctured $X$ with boundary $K$. Note that $γ^0_{S^4}$ is equal to the non-orientable 4-ball genus and hence $γ^0_X$ is a generalization of the non-orientable 4-ball genus. While it is very likely that for given $X$, $γ^0_X$ has no upper bound, it is difficult to show it. In fact, even in the case of $γ^0_{S^4}$, its non-boundedness was shown for the first time by Batson in 2012. In this paper, we prove that fo…

Determining hyperbolicity of compact orientable 3-manifolds with torus boundary

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.

Every orientable 3-manifold is a BΓ

We show that every orientable 3-manifold is a classifying space BΓwhere Γis a groupoid of germs of homeomorphisms of R. This follows by showing that every orientable 3-manifold M admits a codimension one foliation F such that the holonomy cover of every leaf is contractible. The F we construct can be taken to be C^1 but not C^2. The existence of such an F answers positively a question posed by Tsuboi [Classifying spaces for groupoid structures, notes from minicourse at PUC, Rio de Janeiro (2001)], but leaves open the question of whether M = BΓfor some C^\infty groupoid Γ.

Orientable hyperbolic 4-manifolds over the 120-cell

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.

Comparing Heegaard and JSJ structures of orientable 3-manifolds

The Heegaard genus g of an irreducible closed orientable 3-manifold puts a limit on the number and complexity of the pieces that arise in the Jaco-Shalen-Johannson decomposition of the manifold by its canonical tori. For example, if p of the complementary components are not Seifert fibered, then p < g. This result generalizes work of Kobayashi. The Heegaard genus g also puts explicit bounds on the complexity of the Seifert pieces. For example, if the union of the base spaces of the Seifert pieces has Euler characteristic X and there are a total of f exceptional fibers in the Seifert pieces, th…

The Triple Point Spectrum of closed orientable 3-manifolds

The triple point numbers and the triple point spectrum of a closed 3-manifold were defined in (R. Vigara, Representación de 3-variedades por esferas de Dehn rellenantes, PhD Thesis, UNED 2006). They are topological invariants that give a measure of the complexity of a 3-manifold using the number of triple points of minimal filling Dehn surfaces. Basic properties of these invariants are presented, and the triple point spectra of $\mathbb{S}^2\times \mathbb{S}^1$ and $\mathbb{S}^3$ are computed.

Structures of small closed non-orientable 3-manifold triangulations

A census is presented of all closed non-orientable 3-manifold triangulations formed from at most seven tetrahedra satisfying the additional constraints of minimality and P^2-irreducibility. The eight different 3-manifolds represented by these 41 different triangulations are identified and described in detail, with particular attention paid to the recurring combinatorial structures that are shared amongst the different triangulations. Using these recurring structures, the resulting triangulations are generalised to infinite families that allow similar triangulations of additional 3-manifolds to…

Immersions of Surfaces into Aspherical 3-Manifolds

We study finite order invariants of null-homotopic immersions of a closed orientable surface into an aspherical orientable 3-manifold. We give the foundational constructions, and classify all order one invariants.

Embedding closed hyperbolic 3-manifolds in small volume hyperbolic 4-manifolds

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.

不可定向双曲3-流形的形变空间

The deformation space of non-orientable hyperbolic 3-manifolds

本文计算了具有理想三角剖分的有限体积不可定向双曲3-流形的形变空间及其表示簇，发现当端不可定向时存在无法由三角剖分形变实现的表示形变，并在 Gieseking 流形上验证了结论。

On the complexity of non-orientable Seifert fibre spaces

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.

Cosmetic surgeries and non-orientable surfaces

By considering non-orientable surfaces in the surgered manifolds, we show that the 10/3- and -10/3-Dehn surgeries on the 2-bridge knot $9_{27} = S(49,19)$ are not cosmetic, i.e., they give mutually non-homeomorphic manifolds. The knot is unknown to have no cosmetic surgeries by previously known results; in particular, by using the Casson invariant and the Heegaard Floer homology.

3-流形中的纽结与不可定向曲面

Knots and non-orientable surfaces in 3-manifolds

本文提出一种通过嵌入不可定向曲面表示3-流形中纽结与链环的新方法，将其表示为曲面辫群元素的类似于plat的闭包，并在透镜空间和平凡圆丛中给出示例。

Finiteness of mapping degree sets for 3-manifolds

By constructing certain maps, this note completes the answer of the Question: For which closed orientable 3-manifold $N$, the set of mapping degrees $cD(M,N)$ is finite for any closed orientable 3-manifold $M$?

Framings and Projective Framings for 3-Manifolds

We give an elementary proof of the fact that any orientable 3-manifold admits a framing (i.e. is parallelizable) and any non-orientable 3-manifold admits a projective framing. The proof uses only basic facts about immersions of surfaces in 3-space.

On Lagrangian embeddings of closed non-orientable 3-manifolds

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.

Determining the trisection genus of orientable and non-orientable PL 4-manifolds through triangulations

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…

Genus two Heegaard splittings of orientable three-manifolds

It was shown by Bonahon-Otal and Hodgson-Rubinstein that any two genus-one Heegaard splittings of the same 3-manifold (typically a lens space) are isotopic. On the other hand, it was shown by Boileau, Collins and Zieschang that certain Seifert manifolds have distinct genus-two Heegaard splittings. In an earlier paper, we presented a technique for comparing Heegaard splittings of the same manifold and, using this technique, derived the uniqueness theorem for lens space splittings as a simple corollary. Here we use a similar technique to examine, in general, ways in which two non-isotopic genus-…

CR regular embeddings and immersions of compact orientable 4-manifolds into C3

We show that a compact orientable 4-manifold M has a CR regular immersion into C3 if and only if both its first Pontryagin class and its Euler characteristic vanish, and has a CR regular embedding into C3 if and only if in addition the second Stiefel-Whitney class of M vanishes.

Manifolds admitting a continuous cancellative binary operation are orientable

We prove that a topological manifold (possibly with boundary) admitting a continuous cancellative binary operation is orientable. This implies that the Möbius band admits no cancellative continuous binary operation. This answers a question posed by the second author in 2010.

某些 cusps 转移的双曲 4 维流形

Some cusp-transitive hyperbolic 4-manifolds

本文研究了4种闭合可定向平坦3维流形作为双曲4维流形的cusp截面，其对称群在cusp集合上转移。

The 191 orientable octahedral manifolds

We enumerate all spaces obtained by gluing in pairs the faces of the octahedron in an orientation-reversing fashion. Whenever such a gluing gives rise to non-manifold points, we remove small open neighbourhoods of these points, so we actually deal with three-dimensional manifolds with (possibly empty) boundary. There are 298 combinatorially inequivalent gluing patterns, and we show that they define 191 distinct manifolds, of which 132 are hyperbolic and 59 are not. All the 132 hyperbolic manifolds were already considered in different contexts by other authors, and we provide here their known `…

源自着色三角剖分的 PL 4-流形三分剖分

Trisections of PL 4-manifolds arising from colored triangulations

本文将gem-诱导三分剖分推广至不可定向紧4-流形，并证明特定边界条件下可自然导出闭4-流形的三分剖分，进而通过着色三角剖分给出闭可定向4-流形三分剖分亏格的估计。

关于允许共定向紧致叶面但无消失欧拉类的3维流形

On 3-manifolds admitting co-orientable taut foliations, but none with vanishing Euler class

本文构造了允许共定向紧致叶面但无消失欧拉类的3维流形，揭示了其与L空间猜想及左可序基本群的关系。

Embeddings of braid groups into mapping class groups and their homology

We construct several families of embeddings of braid groups into mapping class groups of orientable and non-orientable surfaces and prove that they induce the trivial map in stable homology in the orientable case, but not so in the non-orientable case. We show that these embeddings are non-geometric in the sense that the standard generators of the braid group are not mapped to Dehn twists.

A census of genus two 3-manifolds up to 42 coloured tetrahedra

We improve and extend to the non-orientable case a recent result of Karabas, Malicki and Nedela concerning the classification of all orientable prime 3-manifolds of Heegaard genus two, triangulated with at most 42 coloured tetrahedra.

Incompressible surfaces in handlebodies and boundary reducible 3-manifolds

We study the existence of incompressible embeddings of surfaces into the genus two handlebody. We show that for every compact surface with boundary, orientable or not, there is an incompressible embedding of the surface into the genus two handlebody. In the orientable case the embedding can be either separating or non-separating. We also consider the case in which the genus two handlebody is replaced by an orientable 3-manifold with a compressible boundary component of genus greater than or equal to two.

Right-angled Coxeter polytopes, hyperbolic 6-manifolds, and a problem of Siegel

By gluing together the sides of eight copies of an all-right angled hyperbolic 6-dimensional polytope, two orientable hyperbolic 6-manifolds with Euler characteristic -1 are constructed. They are the first known examples of orientable hyperbolic 6-manifolds having the smallest possible volume.

Stable concordance of knots in 3-manifolds

Knots and links in 3-manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato-Levine invariants, and Milnor's triple linking numbers. Besides fitting into a general theory of Whitney towers, these invariants provide obstructions to the existence of a singular concordance which can be homotoped to an embedding after stabilization by connected sums with $S^2\times S^2$. Results include classifications of stably slice links in orientable 3-manifolds, stable knot concordance in products of an orie…

Minimal 4-colored graphs representing an infinite family of hyperbolic 3-manifolds

The graph complexity of a compact 3-manifold is defined as the minimum order among all 4-colored graphs representing it. Exact calculations of graph complexity have been already performed, through tabulations, for closed orientable manifolds (up to graph complexity 32) and for compact orientable 3-manifolds with toric boundary (up to graph complexity 12) and for infinite families of lens spaces. In this paper we extend to graph complexity 14 the computations for orientable manifolds with toric boundary and we give two-sided bounds for the graph complexity of tetrahedral manifolds. As a consequ…

Sol流形中的Z₂-Thurston规范与非定向曲面的可嵌入性

$\mathbb{Z}_2$-Thurston norm in Sol manifolds and embeddability of non-orientable surfaces

本文研究Sol流形中Z₂-Thurston规范的计算方法，并确定非定向闭曲面在其中的可嵌入性条件。

闭可定向3-流形中纽结与链环的Milnor不变量

Milnor's invariants for knots and links in closed orientable 3-manifolds

该研究将原本针对三维球面$S^3$中链环的Milnor $\bar{\boldsymbol{\nu}}$-不变量推广到一般闭可定向3-流形的纽结与链环场景，证明其为拓扑协边不变量，统一并拓展了三维下的各类Milnor不变量。

Smooth structures on non-orientable four-manifolds and free involutions

In this paper, we investigate existence of inequivalent smooth structures on closed smooth non-orientable 4-manifolds building upon results of Akbulut, Cappell-Shaneson, Fintushel-Stern, Gompf, and Stolz. We add to the number of known constructions and provide new examples of exotic manifolds that are obtained as an application of Gluck twists to the standard smooth structure. Inspection of the smooth structure on the oriented 2-covers yields existence results of orientation-reversing exotic free involutions.