Eiji Ogasa
We introduce a new topological invariant, which is a nonnegative integer, of compact manifolds with boundaries associated with a kind of decomposition of them. Let M and N be m-dimensional compact connected manifolds with boundaries. The new invariant of any boundary union of M and N is less than or equal to that of M (resp. that of N).
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Let $M$ be a triangulated, oriented, connected compact $3$-manifold with connected non-empty boundary. Such a manifold admits a unique decomposition into $\triangle$-prime $3$-manifolds. In this paper, we show that the adjoint Reidemeister torsion has a multiplicative property on the disk sum decomposition of compact $3$-manifolds without a corrective term.
We show that open 3-manifolds that have a locally finite decomposition along 2-spheres are characterized by the existence of a Riemannian metric with respect to which the second homotopy group of the manifold is generated by small elements.
A theorem on the existence of the unique minimal topologic handle decomposition of differentiable simply connected five-dimensional manifolds is proved. For a decomposition of this sort, the number of handles of each index is given.
It has been shown by V. Colin that every tight contact 3-manifold can be written as a connected sum of prime manifolds. Here we prove that the summands in this decomposition are unique up to order and contactomorphism.
One-ended 3-manifolds without locally finite toric decompositions
针对开3维流形的环面分解存在性问题，论文用两个紧致3维流形递归构造一类单端流形，并给出不存在 JSJ 型局部有限环面分解的反例。
The classical Kneser-Milnor theorem says that every closed oriented connected 3-dimensional manifold admits a unique connected sum decomposition into manifolds that cannot be decomposed any further. We discuss to what degree such decompositions exist in higher dimensions and we show that in many settings uniqueness fails in higher dimensions.
In this paper we extend previous results concerning the behaviour of JSJ decompositions of closed 3-manifolds with respect to the profinite completion to the case of compact 3-manifolds with boundary. We also illustrate an alternative and perhaps more natural approach to part of the original theorem, using relative cohomology to analyse the actions of an-annular atoroidal groups on profinite trees.
We give examples of open 3-manifolds and 3-orbifolds that exhibit pathological behavior with respect to splitting along surfaces (2-suborbifolds) with nonnegative Euler characteristic.
The profinite completion of the fundamental group of a closed, orientable $3$-manifold determines the Kneser--Milnor decomposition. If $M$ is irreducible, then the profinite completion determines the Jaco--Shalen--Johannson decomposition of $M$.
In this paper we study colored triangulations of compact PL 4-manifolds with empty or connected boundary which induce handle decompositions lacking in 1-handles or in 1- and 3-handles, thus facing also the problem, posed by Kirby, of the existence of {\it special handlebody decompositions} for any simply-connected closed PL 4-manifold. In particular, we detect a class of compact simply-connected PL 4-manifolds with empty or connected boundary, which admit such decompositions and, therefore, can be represented by (undotted) framed links. Moreover, this class includes any compact simply-connecte…
Handlebody decompositions of 3-manifolds and polycontinuous patterns
本文提出3维流形的柄体分解概念，证明闭可定向3维流形的任意两个柄体分解稳定等价，并将其用于多连续模式建模及嵌段共聚物微相分离的拓扑研究。
A pants-block decomposition of a 3-manifold is similar to a triangulation of a 3-manifold in many aspects. In this paper we show that any two pants-block decompositions of a 3-manifold are related by a finite sequence of moves which are called P-moves. The P-moves between pants-block decompositions are similar to the Pachner moves between triangulations. Moreover, we also give a list of types of P-moves. The main tools we used in this paper are the Morse 2-functions, Reeb complexes and a new 2-dimensional complex called P-complex.
In this paper, we study face vectors of simplicial posets that are the face posets of cell decompositions of topological manifolds without boundary. We characterize all possible face vectors of simplicial posets whose geometric realizations are homeomorphic to the product of spheres. As a corollary, we obtain the characterization of face vectors of simplicial posets whose geometric realizations are odd dimensional manifolds without boundary.
We show that every smooth closed oriented four-manifold admits a decomposition into two co- dimension zero submanifolds with common boundary. Each of these submanifolds carries a structure of a symplectic manifold with pseudo-convex boundary. This imply, in particular, that every smooth closed simply-connected four-manifold is a Stein domain in the the complement of a certain contractible 2-complex.
We describe a new approach to the canonical decompositions of 3-manifolds along tori and annuli due to Jaco-Shalen and Johannson (with ideas from Waldhausen) - the so-called JSJ-decomposition theorem. This approach gives an accessible proof of the decomposition theorem; in particular it does not use the annulus-torus theorems, and the theory of Seifert fibrations does not need to be developed in advance.
We prove a decomposition formula for twisted Blanchfield pairings of 3-manifolds. As an application we show that the twisted Blanchfield pairing of a 3-manifold obtained from a 3-manifold Y with a representation $ϕ: Z[π_1(Y)] \to R$, infected by a knot J along a curve $η$ with $ϕ(η) \neq 1$, splits orthogonally as the sum of the twisted Blanchfield pairing of Y and the ordinary Blanchfield pairing of the knot J, with the latter tensored up from $Z[t,t^{-1}]$ to R.
It is known by A. Loi and R. Piergallini that a closed, oriented, smooth 3-manifold is Stein fillable if and only if it has a positive open book decomposition. In the present paper we will show that for every link L in a Stein fillable 3-manifold there exists an additional knot L' to L such that the union of the links L and L' is the binding of a positive open book decomposition of the Stein fillable 3-manifold. To prove the assertion, we will use the divide, which is a generalization of real morsification theory of complex plane curve singularities, and 2-handle attachings along Legendrian cu…
We show that if $G$ is an upper semicontinuous decomposition of $\mathbb{R}^n$, $n \geq 4$, into convex sets, then the quotient space $\mathbb{R}^n/G$ is a codimension one manifold factor. In particular, we show that $\mathbb{R}^n/G$ has the disjoint arc-disk property.
We try to understand the geometric properties of $n$-manifolds ($n\geq 2$) with geometric structures modeled on $(\bR P^n, \PGL(n+1, \bR))$, i.e., $n$-manifolds with projectively flat torsion free affine connections. We define the notion of $i$-convexity of such manifolds due to Carriére for integers $i$, $1 \leq i \leq n-1$, which are generalization of convexity. Given a real projective $n$-manifold $M$, we show that the failure of an $(n-1)$-convexity of $M$ implies an existence of a certain geometric object, $n$-crescent, in the completion $\che M$ of the universal cover $\tilde M$ of $M$. …
The JSJ-decomposition of the 3-manifold obtained by 0-surgery along a classical pretzel knot of genus one
本文研究沿亏格一经典扭结作0-手术所得3-流形的JSJ分解，并借助最小扭曲五链环异常填充的分类结果进行分析。
The polyhedral decomposition of cusped hyperbolic $n$-manifolds with totally geodesic boundary
该研究针对带全测地边界的体积有限非紧完备双曲$n$-流形，证明其存在由理想多面体或带单个截断面的部分截断多面体构成的多面体分解，用两种方法验证结论并证明分解数量有限，完善了相关流形分解理论。
We provide a complete set of two moves that suffice to relate any two open book decompositions of a given 3-manifold. One of the moves is the usual plumbing with a positive or negative Hopf band, while the other one is a special local version of Harer's twisting, which is presented in two different (but stably equivalent) forms. Our approach relies on 4-dimensional Lefschetz fibrations, and on 3-dimensional contact topology, via the Giroux-Goodman stable equivalence theorem for open book decompositions representing homologous contact structures.
In this paper, we show that a smooth 4-manifold diffeomorphic to a complex hypersurface in $\mathbb{CP}^3 $ of degree $d\geq 5$ can be decomposed as the union of $d(d-4)^2$ copies of 4-dimensional pair-of-pants and certain subsets of K3 surfaces.
We present new explicit decompositions of manifolds via so-called fold maps into lower dimensional spaces. Fold maps form a nice class of so-called generic maps, generalizing Morse functions naturally. To understand the topologies and the differentibale structures of manifolds globally, decomposing manifolds are important and this presents interesting topics and problems on geometry of manifolds. The notion of a Heegaard splitting of a $3$-dimensional closed and connected manifold presents a pioneering study. A $3$-dimensional closed and connected manifold is always decomposed into two copies …
There has been much recent interest into those properties of a 3-manifold determined by the profinite completion of its fundamental group. In this paper we give readily computable criteria specifying precisely when two orientable graph manifold groups have isomorphic profinite completions. Our results also distinguish graph manifolds among the class of all 3-manifolds and give information about the structure of totally hyperbolic manifolds, and give control over the pro-$p$ completion of certain graph manifold groups.
A (flat) affine $3$-manifold is a $3$-manifold with an atlas of charts to an affine space $\mathbb{R}^3$ with transition maps in the affine transformation group $\mathrm{Aff}(\mathbb{R}^3)$. We will show that a connected closed affine $3$-manifold is either an affine Hopf $3$-manifold or decomposes canonically to concave affine submanifolds with incompressible boundary, toral $π$-submanifolds and $2$-convex affine manifolds, each of which is an irreducible $3$-manifold. It follows that if there is no toral $π$-submanifold, then $M$ is prime. Finally, we prove that if a closed affine manifold i…
Nonsmooth manifold decompositions
该研究探讨光滑函数连续选择诱导的光滑流形结构，分析其通有性下的分层与非退化临界点下的局部拓扑性质，并通过4维流形上三函数最大值的分析为三分法理论提供新视角。
Decompositions on manifolds appear in various geometric structures. Necessary and sufficient conditions for quotient spaces of decompositions to be manifolds are widely characterized. We characterize necessary and sufficient conditions to be $k$-manifolds $(k = 1, 2)$, which generalize characterizations in the codimension-$k$ cases for the leaf spaces of foliations, the orbit spaces of group-actions, decomposition spaces of upper semi-continuous decompositions, and leaf class spaces of Riemannian foliation with regular closure. To prove such characterizations, we generalize a characterization …
In a variety of settings we provide a method for decomposing a 3-manifold $M$ into pieces. When the pieces have the appropriate type of hyperbolicity, then the manifold $M$ is hyperbolic and its volume is bounded below by the sum of the appropriately defined hyperbolic volumes of the pieces. A variety of examples of appropriately hyperbolic pieces and volumes are provided, with many examples from link complements in the 3-sphere.
Under mild topological restrictions, this article establishes that a smooth, closed, simply connected manifold of dimension at most seven which can be decomposed as the union of two disk bundles must be rationally elliptic. In dimension five, such manifolds are classified up to diffeomorphism, while the same is true in dimension six when either the second Betti number vanishes or the third Betti number is non-trivial.
We show, up to h-cobordism, that the existence and uniqueness of connected sum decompositions of oriented 4-dimensional manifolds is an invariant of homotopy equivalence, assuming that the fundamental group of each summand is "good" in the sense of Freedman and Quinn. On a separate note, we observe that the Borel Conjecture is true in dimension 4 up to s-cobordism, assuming that the fundamental group satisfies the Farrell--Jones Conjecture.
In this article we show that every closed oriented smooth 4-manifold can be decomposed into two codimension zero submanifolds (one with reversed orientation) so that both pieces are exact Kahler manifolds with strictly pseudoconvex boundaries and that induced contact structures on the common boundary are isotopic. Meanwhile, matching pairs of Lefschetz fibrations with bounded fibers are offered as the geometric counterpart of these structures. We also provide a simple topological proof of the existence of folded symplectic forms on 4-manifolds.
Convex projective manifolds, symmetric spaces and geometric decompositions
研究闭的、不可分解的、恰当凸实射影4-流形，证明其几何分解时每块是实双曲的，推广了定理到四维，还构建特定4-流形，并刻画了支持恰当凸实射影结构的紧致局部对称空间。
The E_8-manifold has several natural framed link descriptions, and we give an efficient method (via `grapes') for showing that they are indeed the same 4-manifold. This leads to explicit handle pictures for the perturbation of singular fibers in an elliptic surface to a collection of fishtails. In the same vein, we show how the degeneration of a regular fiber to a singular fiber in an elliptic surface provides rich examples of Gromov's compactness theorem.
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。
In this paper, we want to discuss the topology of the non-singular hypersurface $Y^{n}$ with complex dimension $n$ in a projective toric manifold $X^{n+1}$. When $n$ is odd, our main results are a decomposition of $Y^{n}\cong Y'\sharp \ s(S^n \times S^n) $ as a connected sum of $s$ copies of $S^n \times S^n$ with a differential manifold $Y'$ such that $b_n (Y')=0$ or 2. When $n$ is even and the degree of $Y$ in $X$ is big enough, we find that $Y$ also admits such a decomposition $Y'\sharp \ s(S^n \times S^n)$, where $Y'$ satisfy $b_n(Y')-|sign(Y')|=b_n(X)\pm sign(H^n(X))$, where $sign(H^n(X))$…
We give a method for obtaining a handle decomposition of an $n$-manifold if the manifold is given by isometric side-pairings of a polyhedron in $\en$, $\sn$ or $\hn$. Every cycle of $k$-faces on the polyhedron corresponds to an $(n-k)$-handle of the manifold. Two applications of the method are given. One helps recognize when a noncompact hyperbolic 3-manifold is a complement of a link in $S^3$ (and automatically produces the link diagram), the other shows that a topological $S^4$ described by the author in \cite{Ivansic3} is diffeomorphic to the standard differentiable $S^4$.
We use convex decomposition theory to (1) reprove the existence of a universally tight contact structure on every irreducible 3-manifold with nonempty boundary, and (2) prove that every toroidal 3-manifold carries infinitely many nonisotopic, nonisomorphic tight contact structures.
Concerns decompositions of smooth 4-manifolds as the union of two handlebodies, each with handles of index <=2 (``Heegard'' decompositions).Sample result: Two 2-complexes are (up to 2-deformation) dual spines of a Heegard decomposition of the 4-sphere if and only if they satisfy the conclusions of the Alexander-Lefshetz duality theorem ($H_1K\simeq H^2L$ and $H_2K\simeq H^1L$).
This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index $\leq2$. Part I gave existence results in terms of spines and chain complexes over the fundamental group of the ambient manifold. Here we assume that one side of a decomposition has larger fundamental group, and use this to define algebraic-topological invariants. These reveal a basic asymmetry in these decompositions: subtle changes on one side can force algebraic-topologically detectable changes on the other. A solvable iteration of the basic invariant gives an ``obstruction th…
The t-invariant can be considered as the Turaev-Viro invariant of order 5 computed for integer colors only. We compute all values of the t-invariant for Seifert manifolds with base sphere and three singular fibers. As a result we show that the manifolds parameters modulo five define the value of the t-invariant. Partially we show that there are 12 distinct values of the t-invariant for these manifolds. Some examples show that the t-invariant for these manifolds is not defined by the first homology group.
In this paper, it is explained that a topological invariant for 3-manifold $M$ with $b_1(M)=1$ can be constructed by applying Fukaya's Morse homotopy theoretic approach for Chern--Simons perturbation theory to a local system on $M$ of rational functions associated to the free abelian covering of $M$. Our invariant takes values in Garoufalidis--Rozansky's space of Jacobi diagrams whose edges are colored by rational functions. It is expected that our invariant gives a lot of nontrivial finite type invariants of 3-manifolds.
Cooper and Long generalised Epstein and Penner's Euclidean cell decomposition of cusped hyperbolic manifolds of finite volume to non-compact strictly convex projective manifolds of finite volume. We show that Weeks' algorithm to compute this decomposition for a hyperbolic surface generalises to strictly convex projective surfaces.
Asymptotic dimension and geometric decompositions in dimensions 3 and 4
本文证明Thurston几何分解的4维流形基本群渐近维数至多4，闭3维流形群至多3，并由此推出Novikov猜想成立及若干几何推论。
We prove that in dimension 3 every nondegenerate contact form is carried by a broken book decomposition. As an application we get that if M is a closed irreducible oriented 3-manifold that is not a graph manifold, for example a hyperbolic manifold, then every nondegenerate Reeb vector field on M has positive topological entropy. Moreover, we obtain that on a closed 3-manifold, every nondegenerate Reeb vector field has either two or infinitely many periodic orbits, and two periodic orbits are possible only on the sphere or on a lens space.
A new class of 3-manifold invariants is constructed from representations of the category of framed tangles.
A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental group of a compact orientable $3$-manifold $M$ has a generalized torsion element if and only if the fundamental group of some prime factor of $M$ has a generalized torsion element. On the other hand, we demonstrate that there are infinitely many toroidal $3$-manifolds whose fund…
We establish a framework for extending invariants of sutured manifolds to invariants of pairs of sutured manifolds who differ by attaching a basic slice along a torus boundary component. In the particular case of (bordered-)sutured Floer homology we show that this recovers the bordered-sutured Floer homology of a corresponding bordered-sutured three-manifold - though the framework is broad enough that it should be applicable in the setting of other invariants of sutured manifolds.
We introduce a generalization of the Dijkgraaf-Witten invariants for cusped or compact oriented 3-manifolds. We show that the generalized DW invariants distinguish some pairs of cusped hyperbolic 3-manifolds with the same hyperbolic volumes and with the same Turaev-Viro invariants. We also present an example of a pair of cusped hyperbolic 3-manifolds with the same hyperbolic volumes and with the same homology groups, whereas with distinct generalized DW invariants.
Motivated by the Turaev-Viro invariant of 3-manifolds, we construct a formal topological invariant of closed, oriented 3-manifolds involving spherical tetrahedra as an application of the asymptotic formula of 6j symbols for the Quantum Enveloping Algebra of sl(2). This invariant can be considered as a spherical version of a formal invariant of Ponzano-Regge and Korepanov defined via Euclidean tetrahedra.
We propose a modification of the three-manifold invariant based on the use of Euclidean metric values ascribed to the elements of manifold triangulation. We thus obtain a nontrivial invariant that can, in particular, distinguish non-homeomorphic lens spaces.
A new quantum gauge model is proposed. From this quantum gauge model we derive a quantum invariant of 3-manifolds. We show that this quantum invariant of 3-manifolds gives a classification of closed (orientable and connected) 3-manifolds. From this classification we then prove the Poincaré conjecture.
We construct an invariant of open four-manifolds using the Heegaard Floer theory of Ozsvath and Szabo. We show that there is a manifold X homeomorphic to R^4 for which the invariant is non-trivial, showing that X is an exotic R^4.
Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in $\mathbb{CP}^{n+1}$ decomposes into pairs of pants: a pair of pants is a real compact $2n$-manifold with cornered boundary obtained by removing an open regular neighborhood of $n+2$ generic hyperplanes from $\mathbb{CP}^n$. As is well-known, every compact surface of genus $g\geqslant 2$ decomposes into pairs of pants, and it is now natural to investigate this construction in dimension 4. Which smooth closed 4-manifolds decompose into pairs of pants? We address this problem here and construct many examples: …
We construct a hyperbolic three-manifold with trivial finite type invariants up to a given degree.
We prove that every quasitoric manifold admits an invariant metric of positive scalar curvature.
If a real value invariant of compact combinatorial manifolds (with or without boundary) depends only on the number of simplices in each dimension on the manifold, then the invariant is completely determined by Euler characteristics of the manifold and its boundary. So essentially, Euler characteristic is the unique invariant of this type.
This is a survey article on finite type invariants of 3-manifolds written for the Encyclopedia of Mathematical Physics to be published by Elsevier.
We calculate Perelman's invariant for compact complex surfaces and a few other smooth four-manifolds. We also prove some results concerning the dependence of Perelman's invariant on the smooth structure.
New estimates are derived concerning the behavior of self-dual hamonic 2-forms on a compact Riemannian 4-manifold with non-trivial Seiberg-Witten invariants. Applications include a vanishing theorem for certain Seiberg-Witten invariants on compact 4-manifolds of constant negative sectional curvature.
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