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1.
We investigate slicings of combinatorial manifolds as properly embedded co-dimension 1 submanifolds. Focus is given to the case of dimension 3, where slicings are (discrete) normal surfaces. For the cases of 2-neighborly 3-manifolds as well as quadrangulated slicings, lower bounds on the number of quadrilaterals of slicings depending on its genus g are presented. These are shown to be sharp for infinitely many values of g. Furthermore, we classify slicings of combinatorial 3-manifolds which are weakly neighborly polyhedral maps.  相似文献   

2.
We study how to realize Smale solenoid type attractors in 3-manifolds. It is already known that we can restrict the 3-manifolds to lens spaces. We get all Smale solenoids realized in a given lens space through an inductive construction. We turn this around to address the question of how to decide whether a closed braid is a trivial knot in S3. For a diffeomorphism f of a 3-manifold M that realizes a Smale solenoid, it is natural to ask whether f−1 also realizes a Smale solenoid. We relate this question to exchangeable braids, and for some special positive case, we describe the relation between the two Smale solenoids of f and f−1.  相似文献   

3.
We make a detailed study of the Heegaard Floer homology of the product of a closed surface Σg of genus g with S1. We determine HF+(Σg×S1,s;C) completely in the case c1(s)=0, which for g?3 was previously unknown. We show that in this case HF is closely related to the cohomology of the total space of a certain circle bundle over the Jacobian torus of Σg, and furthermore that HF+(Σg×S1,s;Z) contains nontrivial 2-torsion whenever g?3 and c1(s)=0. This is the first example known to the authors of torsion in Z-coefficient Heegaard Floer homology. Our methods also give new information on the action of H1(Σg×S1) on HF+(Σg×S1,s) when c1(s) is nonzero.  相似文献   

4.
We prove that an invariant of closed 3-manifolds, called the block number, which is defined via flow-spines, equals the Heegaard genus, except for S 3 and S 2 × S 1. We also show that the underlying 3-manifold is uniquely determined by a neighborhood of the singularity of a flow-spine. This allows us to encode a closed 3-manifold by a sequence of signed labeled symbols. The behavior of the encoding under the connected sum and a criterion for reducibility are studied.  相似文献   

5.
J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631-657] used the curve complex associated to the Heegaard surface of a splitting of a 3-manifold to study its complexity. He introduced the distance of a Heegaard splitting as the distance between two subsets of the curve complex associated to the handlebodies. Inspired by a construction of T. Kobayashi [T. Kobayashi, Casson-Gordon's rectangle condition of Heegaard diagrams and incompressible tori in 3-manifolds, Osaka J. Math. 25 (3) (1988) 553-573], J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631-657] proved the existence of arbitrarily high distance Heegaard splittings.In this work we explicitly define an infinite sequence of 3-manifolds {Mn} via their representative Heegaard diagrams by iterating a 2-fold Dehn twist operator. Using purely combinatorial techniques we are able to prove that the distance of the Heegaard splitting of Mn is at least n.Moreover, we show that π1(Mn) surjects onto π1(Mn−1). Hence, if we assume that M0 has nontrivial boundary then it follows that the first Betti number β1(Mn)>0 for all n?1. Therefore, the sequence {Mn} consists of Haken 3-manifolds for n?1 and hyperbolizable 3-manifolds for n?3.  相似文献   

6.
Let K be a knot in a sphere S3. We denote by t(K) the tunnel number of K. For two knots K1 and K2, we denote by K1?K2 the connected sum of K1 and K2. In this paper, we will prove that if one of K1 and K2 has high distance while the other has distance at least 3 then t(K1?K2)=t(K1)+t(K2)+1.  相似文献   

7.
We compute the p-primary components of the linking pairings of orientable 3-manifolds admitting a fixed-point free S1-action. Any linking pairing on a finite abelian group of odd order is realized by such a manifold. We find necessary and sufficient conditions for a pairing on an abelian 2-group to be the 2-primary component of such a linking pairing, and give simple examples which are not realizable by any Seifert fibred 3-manifold.  相似文献   

8.
We deal with Riemannian properties of the octonionic Hopf fibration S 15S 8, in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the non-existence of S 1 subfibrations. We then discuss Spin(9)-structures from a conformal viewpoint and determine the structure of compact locally conformally parallel Spin(9)-manifolds. Eventually, we give a list of examples of locally conformally parallel Spin(9)-manifolds.  相似文献   

9.
In the present paper, we consider a class of compact orientable 3-manifolds with one boundary component, and suppose that the manifolds are ?-reducible and admit complete surface systems. One of our main results says that for a compact orientable, irreducible and ?-reducible 3-manifold M with one boundary component F of genus n > 0 which admits a complete surface system S′, if D is a collection of pairwise disjoint compression disks for ?M , then there exists a complete surface system S for M , which is equivalent to S′, such that D is disjoint from S . We also obtain some properties of such 3-manifolds which can be embedded in S3.  相似文献   

10.
By means of a slight modification of the notion of GM-complexity introduced in [Casali, M.R., Topol. Its Appl., 144: 201–209, 2004], the present paper performs a graph-theoretical approach to the computation of (Matveev’s) complexity for closed orientable 3-manifolds. In particular, the existing crystallization catalogue available in [Lins, S., Knots and Everything 5, World Scientific, Singapore, 1995] is used to obtain upper bounds for the complexity of closed orientable 3-manifolds triangulated by at most 28 tetrahedra. The experimental results actually coincide with the exact values of complexity, for all but three elements. Moreover, in the case of at most 26 tetrahedra, the exact value of the complexity is shown to be always directly computable via crystallization theory.  相似文献   

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