This paper presents some finiteness results for the number of boundary slopes of immersed essential surfaces of given genus g in a compact 3-manifold with torus boundary. In the case of hyperbolic 3-manifolds we obtain uniform quadratic bounds in g for the number of possible slopes, independent of the 3-manifold. We also look at some related quantities, such as how many times the slopes of two such surfaces of specified genus can intersect.
For any hyperbolic 3-manifold $M$ with totally geodesic boundary, there are finitely many boundary slopes for essential immersed surfaces of a given genus. There is a uniform bound for the number of such boundary slopes if the genus of $\partial M$ or the volume of $M$ is bounded above. When the volume is bounded above, then area of $\partial M$ is bounded above and the length of closed geodesic on $\partial M$ is bounded below.
Let $F$ be a proper essential immersed surface in a hyperbolic 3-manifold $M$ with boundary disjoint from a torus boundary component $T$ of $M$. Let $α$ be the set of coannular slopes of $F$ on $T$. The main theorem of the paper shows that there is a constant $K$ and a finite set of slopes $Λ$ on $T$, such that if $β$ is a slope on $T$ with $Δ(β, α_i) > K$ for all $α_i$ in $α$, and $β$ is not in $Λ$, then $F$ remains incompressible after Dehn filling on $T$ along the slope $β$. In certain sense, this means that $F$ survives most Dehn fillings. The proof uses minimal surface theory, integral of…
We show that an immersed thrice-punctured sphere in a cusped orientable hyperbolic 3-manifold is either embedded or has a single clasp in a manifold obtained by hyperbolic Dehn filling on a cusp of the Whitehead link complement.
In this paper, we exploit a subtle indeterminacy in the definition of the spherical Kervaire-Milnor invariant which was discovered by R. Stong to construct non-spin 4-manifolds with even intersection form and prescribed signature.
We show that for certain hyperbolic 3-manifolds, all boundary slopes are slopes of immersed incompressible surfaces, covered by incompressible embeddings in some finite cover. The manifolds include hyperbolic punctured torus bundles and hyperbolic two-bridge knots.
We show that many 3-manifold groups have no nonabelian surface subgroups. For example, any link of an isolated complex surface singularity has this property. In fact, we determine the exact class of closed graph-manifolds which have no immersed pi_1-injective surface of negative Euler characteristic. We also determine the class of closed graph manifolds which have no finite cover containing an embedded such surface. This is a larger class. Thus, manifolds M^3 exist which have immersed pi_1-injective surfaces of negative Euler characteristic, but no such surface is virtually embedded (finitely …
Let $S_0$ and $S_1$ be two homotopic, oriented 2-spheres embedded in an orientable 4-manifold $X$. After discussing several operations for modifying an immersion of a 3-manifold into a 5-manifold, we discuss the Freedman--Quinn (fq) and Stong (stong) concordance obstructions. When these are defined for the pair $S_0,S_1$, they are defined in terms of the self-intersection set of a regular homotopy from $S_0$ to $S_1$. When $S_0$ has an immersed dual sphere, we see that under some mild topological conditions on $X$, the invariants fq and stong are a complete set of concordance obstructions. Thi…
We define an invariant, which we call surface-complexity, of closed 3-manifolds by means of Dehn surfaces. The surface-complexity of a manifold is a natural number measuring how much the manifold is complicated. We prove that it fulfils interesting properties: it is subadditive under connected sum and finite-to-one on P2-irreducible manifolds. Moreover, for P2-irreducible manifolds, it equals the minimal number of cubes in a cubulation of the manifold, except for the sphere S3, the projective space RP3 and the lens space L41, which have surface-complexity zero. We will also give estimations of…
Let $M$ be an orientable 3-manifold with $\partial M$ a single torus. We show that the number of boundary slopes of immersed essential surfaces with genus at most $g$ is bounded by a quadratic function of $g$. In the hyperbolic case, this was proved earlier by Hass, Rubinstein and Wang.
The paper contains a new proof that a complete, non-compact hyperbolic $3$-manifold $M$ with finite volume contains an immersed, closed, quasi-Fuchsian surface.
Among other things, we prove the following two topologcal statements about closed hyperbolic 3-manifolds. First, every rational second homology class of a closed hyperbolic 3-manifold has a positve integral multiple represented by an oriented connected closed $π_1$-injectively immersed quasi-Fuchsian subsurface. Second, every rationally null-homologous, $π_1$-injectively immersed oriented closed 1-submanifold in a closed hyperbolic 3-manifold has an equidegree finite cover which bounds an oriented compact $π_1$-injective immersed quasi-Fuchsian subsurface. In part, we exploit techniques develo…
In this note, we investigate the relation between double points and complex points of immersed surfaces in almost-complex 4-manifolds and show how estimates for the minimal genus of embedded surfaces lead to inequalities between the number of double points and the number of complex points of an immersion. We also provide a generalization of a classical genus estimate due to V.A. Rokhlin to the case of immersed surfaces.
We count the number of conjugacy classes of maximal, genus g, surface subroups in hyperbolic 3-manifold groups. For any closed hyperbolic 3-manifold, we show that there is an upper bound on this number which grows factorially with g. We also give a class of closed hyperbolic 3-manifolds for which there is a lower bound of the same type.
Suppose $δ$ is a generic immersed closed curve in the boundary of a 3-manifold M and $δ$ is null-homotopic in M. Then $δ$ can be displaced by a height function in a collar of the boundary so that the resulting simple closed curve in the collar bounds a disk in M.
We generalize the definition of thin position of Scharlemann and Thompson for compact orientable 3-manifolds with torus boundary components and introduce $α$-sloped generalized Heegaard splittings. We examine its relationship to generalized Heegaard splittings of manifolds resulting from Dehn filling. We compare alpha-sloped thin position of 3-manifolds to other types of thin position for knots and 3-manifolds and discuss how this kind of decomposition gives an organic picture of M and allows the structure of the manifold to dictate the most natural slope(s) on the boundary. Additionally, we p…
A combinatorial condition is obtained for when immersed or embedded incompressible surfaces in compact 3-manifolds with tori boundary components remain incompressible after Dehn surgery. A combinatorial characterisation of hierarchies is described. A new proof is given of the topological rigidity theorem of Hass and Scott for 3-manifolds containing immersed incompressible surfaces, as found in cubings of non-positive curvature.
The paper introduces the spirality character of the almost fiber part for a closed essentially immersed subsurface of a closed orientable aspherical 3-manifold, which generalizes an invariant due to Rubinstein and Wang. The subsurface is virtually embedded if and only if the almost fiber part is aspiral, and in this case, the subsurface is virtually a leaf of a taut foliation. Besides other consequences, examples are exhibited that non-geometric 3-manifolds with no Seifert fibered pieces may contain essentially immersed but not virtually embedded closed subsurfaces.
Motivated by the L-space conjecture, we investigate various notions of order-detection of slopes on knot manifolds. These notions are designed to characterise when rational homology 3-spheres obtained by gluing compact manifolds along torus boundary components have left-orderable fundamental groups and when a Dehn filling of a knot manifold has a left-orderable fundamental group. Our developments parallel existing results in the case of Heegaard-Floer slope detection and foliation slope detection, leading to several conjectured structure theorems that connect relative Heegaard-Floer homology a…
We show that every closed, virtually fibered hyperbolic 3-manifold contains immersed, quasi-Fuchsian surfaces with convex cores of arbitrarily large thickness.
It has been observed that most manifolds in the Callahan-Hildebrand-Weeks census of cusped hyperbolic $3$-manifolds are obtained by surgery on the minimally twisted 5-chain link. A full classification of the exceptional surgeries on the 5-chain link has recently been completed. In this article, we provide a complete classification of the sets of exceptional slopes and fillings for all cusped hyperbolic surgeries on the minimally twisted 5-chain link, thereby describing the sets of exceptional slopes and fillings for most hyperbolic manifolds of small complexity. The classification produces the…
This paper is to study the $\mathbb{R}^{3}$ case of \hyperref[Zhao]{[9]}. We determine all equivalence classes of immersed $3$-manifolds bounded by an arbitrary immersed surface in $\mathbb{R}^{3}$.
In this paper, we will compute the dimension of the space of spun and ordinary normal surfaces in an ideal triangulation of the interior of a compact 3-manifold with incompressible tori or Klein bottle components. Spun normal surfaces have been described in unpublished work of Thurston. We also define a boundary map from spun normal surface theory to the homology classes of boundary loops of the 3-manifold and prove the boundary map has image of finite index. Spun normal surfaces give a natural way of representing properly embedded and immersed essential surfaces in a 3-manifold with tori and …
A graph manifold rational homology $3$-sphere $W$ with a left-orderable fundamental group admits a co-oriented taut foliation, though it is unknown whether it admits a smooth co-oriented taut foliation. In this paper we extend the gluing theorem of arXiv:1401.7726 to graph manifold rational homology solid tori and use this to show that there are smooth foliations on the pieces of $W$ which come close to matching up on its JSJ tori. This is applied to prove that a graph manifold with left-orderable fundamental group is not an L-space.
We show that the number of double points of smoothly immersed 2-spheres representing certain homology classes of an oriented, smooth, closed, simply-connected 4-manifold X must increase with the complexity of corresponding h-cobordisms from X to X. As an application, we give results restricting the minimal number of double points of immersed spheres in manifolds homeomorphic to rational surfaces.
In this paper we study exceptional Dehn fillings on hyperbolic knot manifolds which contain an essential once-punctured torus. Let $M$ be such a knot manifold and let $β$ be the boundary slope of such an essential once-punctured torus. We prove that if Dehn filling $M$ with slope $α$ produces a Seifert fibred manifold, then $Δ(α,β)\leq 5$. Furthermore we classify the triples $(M; α,β)$ when $\D(α,β)\geq 4$. More precisely, when $\D(α,β)=5$, then $M$ is the (unique) manifold $Wh(-3/2)$ obtained by Dehn filling one boundary component of the Whitehead link exterior with slope -3/2, and $(α, β)$ i…
We give new bounds for the distance between two exceptional filling slopes for a 1-cusped hyperbolic 3-manifold in several different situations. The distance between a reducible slope and a slope that produces a manifold with finite fundamental group is at most 2. The distance between a reducible slope and one that produces a very small manifold is also at most 2. The distance between a reducible slope and one which produces a manifold with a pi_1 injective torus is at most 4. The methods used involve both characteristic submanifold theory and the theory of the PSL(2,C) character variety.
We study the 3-dimensional immersed crosscap number of a knot, which is a nonorientable analogue of the immersed Seifert genus. We study knots with immersed crosscap number 1, and show that a knot has immersed crosscap number 1 if and only if it is a nonntrivial $(2p,q)$-torus or $(2p,q)$-cable knot. We show that unlike in the orientable case the immersed crosscap number can differ from the embedded crosscap number by arbitrarily large amounts, and that it is neither bounded below nor above by the 4-dimensional crosscap number. We then use these constructions to find, for any $n\geq 2$, an ori…
Colorado State University Pueblo（科罗拉多州立大学普韦布洛分校）；Saginaw Valley State University（萨吉诺谷州立大学）；Louisiana State University（路易斯安那州立大学）；Simons Center for Geometry and Physics, State University of New York（纽约州立大学西蒙斯几何与物理中心）
It is shown that given any link-manifold, there is an algorithm to decide if the manifold contains an embedded, essential planar surface; if it does, the algorithm will construct one. If a slope on the boundary of the link-manifold is given, there is an algorithm to determine if the slope bounds an embedded punctured-disk; if a meridian slope is given, then it can be determined if a longitude bounds an embedded punctured-disk. The methods use normal surface theory but do not follow the classical approach. Properties of minimal vertex triangulations, layered-triangulations, 0--efficient triangu…
Let $M$ be a simple 3-manifold such that one component of $\partial M$, say $F$, has genus at least two. For a slope $α$ on $F$, we denote by $M(α)$ the manifold obtained by attaching a 2-handle to $M$ along a regular neighborhood of $α$ on $F$. If $M(α)$ is reducible, then $α$ is called a reducing slope. In this paper, we shall prove that the distance between two separating, reducing slopes on $F$ is at most 4.
We prove that a sequence of possibly branched, weak immersions of the two-sphere $S^2$ into an arbitrary compact riemannian manifold $(M^m,h)$ with uniformly bounded area and uniformly bounded $L^2-$norm of the second fundamental form either collapse to a point or weakly converges as current, modulo extraction of a subsequence, to a Lipschitz mapping of $S^2$ and whose image is made of a connected union of finitely many, possibly branched, weak immersions of $S^2$ with finite total curvature. We prove moreover that if the sequence belongs to a class $γ$ of $π_2(M^m)$ the limiting lipschitz map…
Normal surface theory, a tool to represent surfaces in a triangulated 3-manifold combinatorially, is ubiquitous in computational 3-manifold theory. In this paper, we investigate a relaxed notion of normal surfaces where we remove the quadrilateral conditions. This yields normal surfaces that are no longer embedded. We prove that it is NP-hard to decide whether such a surface is immersed. Our proof uses a reduction from Boolean constraint satisfaction problems where every variable appears in at most two clauses, using a classification theorem of Feder. We also investigate variants, and provide …
Motivated by classical theorems on minimal surface theory in compact hyperbolic three-manifolds, we investigate the questions of existence and deformations for least area minimal surfaces in complete noncompact hyperbolic three-manifold of finite volume. We prove any closed immersed incompressible surface can be deformed to a closed immersed least area surface within its homotopy class in any cusped hyperbolic three-manifold. Our techniques highlight how special structures of these cusped hyperbolic three-manifolds prevent any least area minimal surface going too deep into the cusped region.
We show that, for any given 3-manifold M, there are at most finitely many hyperbolic knots K in the 3-sphere and fractions p/q (with q > 22), such that M is obtained by p/q surgery along K. This is a corollary of the following result. If M is obtained by Dehn filling the cusps of a hyperbolic 3-manifold X, where each filling slope has length more than 2 π+ ε, then, for any given M and ε> 0, there are only finitely many possibilities for X and for the filling slopes. In this paper, we also investigate the length of boundary slopes, and sequences of negatively curved metrics on a given 3-manifol…
The simple loop conjecture for 3-manifolds states that every 2-sided immersion of a closed surface into a 3-manifold is either injective on fundamental groups or admits a compression. This can be viewed as a generalization of the Loop Theorem to immersed surfaces. We prove the conjecture in the case that the target 3-manifold admits a geometric structure modeled on Sol.
We study totally geodesic planes in hyperbolic 3-manifolds $M$ having incompressible core and degenerate ends. We prove a Ratner-type phenomenon: a closed minimal $PSL(2,R)-$invariant subset of $M$ is either an immersed totally geodesic surface or all of $M$. We also show that for an arbitrary infinite volume hyperbolic 3-manifold $M$ without parabolics and with finitely generated fundamental group, the number of compact totally geodesic surfaces in $M$ is finite.
Let $M$ be a simple manifold, and $F$ be a component of $\partial M$ of genus two. For a slope $γ$ on $F$, we denote by $M(γ)$ the manifold obtained by attaching a 2-handle to $M$ along a regular neighborhood of $γ$ on $F$. In this paper, we shall prove that there is at most one separating slope $γ$ on $F$ so that $M(γ)$ is $\partial$-reducible.
We show that the properties of admitting a co-oriented taut foliation and having a left-orderable fundamental group are equivalent for rational homology $3$-sphere graph manifolds and relate them to the property of not being a Heegaard-Floer L-space. This is accomplished in several steps. First we show how to detect families of slopes on the boundary of a Seifert fibred manifold in four different fashions - using representations, using left-orders, using foliations, and using Heegaard-Floer homology. Then we show that each method of detection determines the same family of detected slopes. Next…
We show that every hyperbolic link complement contains closed quasi-Fuchsian surfaces. As a consequence, we obtain the result that on a hyperbolic link complement, if we remove from each cusp of the manifold a certain finite set of slopes, then all remaining Dehn fillings on the link complement yield manifolds with closed immersed incompressible surfaces.
We show that if a hyperbolic knot manifold $M$ contains an essential twice-punctured torus $F$ with boundary slope $β$ and admits a filling with slope $α$ producing a Seifert fibred space, then the distance between the slopes $α$ and $β$ is less than or equal to $5$ unless $M$ is the exterior of the figure eight knot. The result is sharp; the bound of $5$ can be realized on infinitely many hyperbolic knot manifolds. We also determine distance bounds in the case that the fundamental group of the $α$-filling contains no non-abelian free group. The proofs are divided into the four cases $F$ is a …
Given $M_φ$, a fibered 3-manifold with boundary, we show that the translation distance of the monodromy $φ$ can be bounded above by the complexity of an essential surface with non-zero slope. Furthermore we prove that the minimal complexity of a surface with non-zero slope in $M_{φ^n}$ tends to infinity as $n\to\infty$. Additionally, we show that an infinite family of fibered hyperbolic knots has translation distance bounded above by two, satisfying a conjecture by Schleimer which postulates that this behavior should hold for all fibered knots.
The A-polynomial of a manifold whose boundary consists of a single torus is generalised to an eigenvalue variety of a manifold whose boundary consists of a finite number of tori, and the set of strongly detected boundary curves is determined by Bergman's logarithmic limit set, which describes the exponential behaviour of the eigenvalue variety at infinity. This enables one to read off the detected boundary curves of a multi-cusped manifold in a similar way to the 1-cusped case, where the slopes are encoded in the Newton polygon of the A-polynomial.
Let S be an immersed horizontal surface in a 3-dimensional graph manifold. We show that the fundamental group of the surface S is quadratically distorted whenever the surface is virtually embedded (i.e., separable) and is exponentially distorted when the surface is not virtually embedded.
We present a quantitative isolation property of the lifts of properly immersed geodesic planes in the frame bundle of a geometrically finite hyperbolic $3$-manifold. Our estimates are polynomials in the tight areas and Bowen-Margulis-Sullivan densities of geodesic planes, with degree given by the modified critical exponents.
