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Let $ M $ be a compact connected orientable 3-manifold and $ F $ be a compact connected orientable surface properly embedded in $ M $. If $ F $ cuts $ M $ into two handlebodies $ X $ and $ Y $ (i.e., $ M = X\cup_FY $), then we say that $ F $ is an $ H' $-splitting surface for $ M $ and call $ X\cup_FY $ an $ H' $-splitting for $ M $. When the $ H' $-splitting surface $ F $ is incompressible in a handlebody $ H $, a characteristic of an $ H' $-splitting $ H_1\cup_F H_2 $ to denote $ H $ is already known. In the present paper, we generalize the above result as follows: Let $ H $ be a handlebody of genus $ g\geq 1 $, $ X\cup_F Y $ an $ H' $-splitting for $ H $. Then, either $ X\cup_F Y $ is stabilized, or there exists a reducing system $ \mathcal{J}_1\cup\mathcal{K}_1 $ of $ F $, such that $ \mathcal{J}_1 $ is quasi-primitive in $ Y $ and $ \mathcal{K}_1 $ is quasi-primitive in $ X $. Combining the result with the known result, we obtain a characteristic of an $ H' $-splitting $ H_1\cup_F H_2 $ to denote a handlebody.

Let M be a compact connected orientable 3-manifold and F be a compact connected orientable surface properly embedded in M. If F cuts M into two handlebodies X and Y (i.e., M = X ∪F Y), then we say that F is an H'-splitting surface for M and call X ∪F Y an H'-splitting for M. When the H'-splitting surface F is incompressible in a handlebody H, a characteristic of an H'-splitting H1 ∪F H2 to denote H is already known. In the present paper, we generalize the above result as follows: Let H be a handlebody of genus g ≥ 1, X ∪F Y an H'-splitting for H. Then, either X∪F Y is stabilized, or there exists a reducing system J1∪K1 of F, such that J1 is quasi-primitive in Y and K1 is quasi-primitive in X. Combining the result with the known result, we obtain a characteristic of an H'-splitting H1∪F H2 to denote a handlebody [ABSTRACT FROM AUTHOR]

In this paper we study isotopy classes of closed connected orientable surfaces in the standard 3 -sphere. Such a surface splits the 3 -sphere into two compact connected submanifolds, and by using their Heegaard splittings, we obtain a 2 -component handlebody-link. In this paper, we first show that the equivalence class of such a 2-component handlebody-link up to attaching trivial 1 -handles can recover the original surface. Therefore, we can reduce the study of surfaces in the 3 -sphere to that of 2 -component handlebody-links up to stabilizations. Then, by using G -families of quandles, we construct invariants of 2 -component handlebody-links up to attaching trivial 1 -handles, which lead to invariants of surfaces in the 3 -sphere. In order to see the effectiveness of our invariants, we will also show that our invariants can distinguish certain explicit surfaces in the 3 -sphere. [ABSTRACT FROM AUTHOR]

The goal of this paper is to construct distinct trisections of the same genus on a fixed 4-manifold. For every k ≥ 2 , we exhibit infinitely many manifolds with 2 k - 1 non-diffeomorphic (3k, k)-trisections. Here, the manifolds are spun Seifert fiber spaces and the trisections come from Meier's spun trisections. The technique used to distinguish the trisections parallels a common setup for distinguishing Heegaard splittings. In particular, we show that the Nielsen classes of the generators of the fundamental group, obtained from spines of the 4-dimensional 1-handlebodies of the trisection, are isotopy invariants of the trisection. If we additionally consider the action of the automorphism group on the Nielsen classes we obtain diffeomorphism invariants. Once the invariance is established, we analyze the Nielsen classes of the spun trisections in terms of the Nielsen classes of the original Heegaard splitting, and leverage work of Lustig, Moriah, and Rosenberger on Nielsen classes of Fuchsian groups in order to distinguish the trisections. [ABSTRACT FROM AUTHOR]

In this paper, we consider decompositions of 3-manifolds with three handlebodies. We classify such decompositions of the 3-sphere and lens spaces with small genera. These decompositions admit operations called stabilizations. We also determine whether these decompositions are obtained by stabilization from another decomposition. [ABSTRACT FROM AUTHOR]

Given a Heegaard splitting of a three-manifold Y, we consider the SL.2;C/character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies. We focus on the smooth open subset corresponding to irreducible representations. On that subset, the intersection of the Lagrangians is an oriented d-critical locus in the sense of Joyce. Bussi associates to such an intersection a perverse sheaf of vanishing cycles. We prove that in our setting, the perverse sheaf is an invariant of Y, i.e., it is independent of the Heegaard splitting. The hypercohomology of the perverse sheaf can be viewed as a model for (the dual of) SL.2;C/instanton Floer homology. We also present a framed version of this construction, which takes into account reducible representations. We give explicit computations for lens spaces and Brieskorn spheres, and discuss the connection to the Kapustin-Witten equations and Khovanov homology. [ABSTRACT FROM AUTHOR]

In the paper, we give an equivalent description of the lens space L (p , q) with p prime in terms of any corresponding Heegaard diagrams as follows: Let M be a closed orientable 3-manifold with π 1 (M) ≠ 1 , and U ∪ F V a Heegaard splitting of genus n for M with an associated Heegaard diagram (U ; J 1 , ... , J n). Assume p is a prime integer. Then M is homeomorphic to the lens space L (p , q) if and only if there exists an embedding f : U ↪ L (p , q) such that L = { f (J 1) , ... , f (J n) } bounds a complete system of surfaces for W = L (p , q) \ f (U) ¯. [ABSTRACT FROM AUTHOR]

We give a simple criterion for a Heegaard splitting to yield a Haken manifold. As a consequence, we construct many Haken manifolds, in particular homology spheres, with prescribed properties, namely Heegaard genus, Heegaard distance and Casson invariant. Along the way we give simpler and shorter proofs of the existence of splittings with specified Heegaard distance, originally proven by Ido–Jang–Kobayashi, of the existence of hyperbolic manifolds with prescribed Casson invariant, originally due to Lubotzky–Maher–Wu, and of a result about subsurface projections of disc sets (for which we even get better constants), originally due to Masur–Schleimer. [ABSTRACT FROM AUTHOR]

In this paper, we define the minimality of a partition for a critical Heegaard surface. The standard minimal genus Heegaard surface of (a closed orientable surface) × S 1 , which is known to be critical, admits a minimal partition. Moreover, we give an example of a critical surface that admits both a minimal partition and a non-minimal partition. [ABSTRACT FROM AUTHOR]

In this paper, by putting a separating incompressible surface in a 3-manifold into Morse position relative to the height function associated to a strongly irreducible Heegaard splitting, we show that an incompressible subsurface of the Heegaard splitting can be found, by decomposing the 3-manifold along the separating surface. Further if the Heegaard surface is of Hempel distance at least 4, then there is a pair of such subsurfaces on both sides of the given separating surface. This gives a particularly simple hierarchy for the 3-manifold. [ABSTRACT FROM AUTHOR]

We study relations between the structure of the set of equilibrium points of a gradient-like flow and the topology of the support manifold of dimension 4 and higher. We introduce a class of manifolds that admit a generalized Heegaard splitting. We consider gradient-like flows such that the non-wandering set consists of exactly μ node and ν saddle equilibrium points of indices equal to either 1 or n — 1. We show that, for such a flow, there exists a generalized Heegaard splitting of the support manifold of genius g = ν − μ + 2 2 . We also suggest an algorithm for constructing gradientlike flows on closed manifolds of dimension 3 and higher with prescribed numbers of node and saddle equilibrium points of prescribed indices. [ABSTRACT FROM AUTHOR]

In this paper, we consider decompositions of closed orientable 3-manifolds with more than 3 handlebodies, where the union of intersections of handlebodies is a multibranched surface. We define stabilization operations for such decompositions and show the stable equivalence. [ABSTRACT FROM AUTHOR]

We summarize and expand known connections between the study of Dehn surgery on links and the study of trisections of closed, smooth 4-manifolds. In particular, we propose a program in which trisections could be used to disprove the generalized property R conjecture, including a process that converts the potential counterexamples of Gompf, Scharlemann, and Thompson into genus four trisections of the standard 4-sphere that are unlikely to be standard.We also give an analog of the Casson-Gordon rectangle condition for trisections that obstructs reducibility of a given trisection. [ABSTRACT FROM AUTHOR]

Let be a Heegaard splitting of compact orientable -manifold, and be an amalgamation along an essential surface . In this paper, we show that if there is an essential disk in , such that cuts into two product -bundles, one of which is , and , then is unstabilized and is uncritical. We also give a sufficient condition for to be critical. [ABSTRACT FROM AUTHOR]

A -component non-splittable spatial graph in possibly admits a smooth involution on its exterior whose fixed point set is a closed surface of genus . It is known that holds if is a graph link in . In this paper, we consider to be a hyperbolic spatial graph in a closed orientable 3-manifold . We first study the case where is the 3-sphere, and provide a method for constructing and with , where is the Euler characteristic of for . Next, we study the case where is a closed orientable 3-manifold in relation to Heegaard splittings and Dehn surgeries. [ABSTRACT FROM AUTHOR]

Let ( V , W ; F ) be a weakly reducible, unstabilized, Heegaard splitting of genus at least three in an orientable, irreducible 3-manifold M . Then M o d ( M , F ) naturally acts on the disk complex D ( F ) as a group action. In this article, we prove if F is topologically minimal and its topological index is two, then the orbit of any element of D ( F ) for this group action consists of infinitely many elements. Moreover, we prove there are at most two elements of D ( F ) whose orbits are finite if the genus of F is three. [ABSTRACT FROM AUTHOR]

Given a Heegaard splitting of the 3-sphere, the primitive disk complex is defined to be the full subcomplex of the disk complex for one of the handlebodies of the splitting, spanned by the vertices of primitive disks. It is an open question whether the primitive disk complex is connected or not when the genus of the splitting is greater than three. In this note, we prove that a quotient of the primitive disk complex, called the homotopy primitive disk complex, is connected. [ABSTRACT FROM AUTHOR]

The intersecting kernel of a Heegaard splitting for a compact orientable 3-manifold is the subgroup of , where is the homomorphism induced by the inclusion , . In the paper, we obtain some invariants of 3-manifolds from certain quotient groups of the intersecting kernels of their Heegaard splittings. We also list two algebraic problems related to the new invariants, which might be interesting to study. [ABSTRACT FROM AUTHOR]

Let be a -manifold, be an essential planar surface which cuts into two 3-manifolds and . Suppose is a Heegaard splitting of , is the dual Heegaard splitting of and along , where and . In this paper, we give a condition of unstabilized dual Heegaard splittings of -manifolds by using Hempel's distance and the method of proof of Gordon's Conjecture. Also, we give a counterexample for stabilized dual Heegaard splittings of -manifolds for any Hempel's distance. [ABSTRACT FROM AUTHOR]

Let $$M=H_1\cup _S H_2$$ be a Heegaard splitting of a closed orientable 3-manifold M (or a bridge decomposition of a link exterior). Consider the subgroup $${\text {MCG}}^0(H_j)$$ of the mapping class group of $$H_j$$ consisting of mapping classes represented by orientation-preserving auto-homeomorphisms of $$H_j$$ homotopic to the identity, and let $$G_j$$ be the subgroup of the automorphism group of the curve complex $$\mathcal {CC}(S)$$ obtained as the image of $${\text {MCG}}^0(H_j)$$ . Then the group $$G=\langle G_1, G_2\rangle $$ generated by $$G_1$$ and $$G_2$$ acts on $$\mathcal {CC}(S)$$ with each orbit being contained in a homotopy class in M. In this paper, we study the structure of the group G and examine whether a homotopy class can contain more than one orbit. We also show that the action of G on the projective lamination space of S has a non-empty domain of discontinuity when the Heegaard splitting satisfies R-bounded combinatorics and has high Hempel distance. [ABSTRACT FROM AUTHOR]

In this work, we first present a necessary and sufficient condition to closed orientable 3-manifold admitting keen weakly reducible Heegaard splitting. By an example, we show that condition is not suitable for compact orientable manifold with non-sphere boundary component. At last, we give a necessary and sufficient condition for compact orientable 3-manifold to have a keen weakly reducible Heegaard splitting. [ABSTRACT FROM AUTHOR]