Classifying foliations of 3-manifolds via branched surfaces

We use branched surfaces to define an equivalence relation on C¹ codimension one foliations of any closed orientable 3-manifold that are transverse to some fixed nonsingular flow. There is a discrete metric on the set of equivalence classes with the property that foliations that are sufficiently close (up to equivalence) share important topological properties.     

On the height of 2-component links

Let p be a prime and let L be a 2-component link in S³. We define a numerical invariant, called p-height of L, using a tower of successive p-fold branched cyclic coverings of L, and show, in particular, 2-height is algorithmically determined for any 2-component link. Some relationships between p-height and known link type invariants are also established.     

Estimating Matveev's complexity via crystallization theory

In [M.R. Casali, Computing Matveev's complexity of non-orientable 3-manifolds via crystallization theory, Topology Appl. 144(1-3) (2004) 201-209], a graph-theoretical approach to Matveev's complexity computation is introduced, yielding the complete classification of closed non-orientable 3-manifolds up to complexity six. The present paper follows the same point-of view, making use of crystallization theory and related results (see [M. Ferri, Crystallisations of 2-fold branched coverings of S³, Proc. Amer. Math. Soc. 73 (1979) 271-276; M.R. Casali, Coloured knots and coloured graphs representing 3-fold simple coverings of S³, Discrete Math. 137 (1995) 87-98; M.R. Casali, From framed links to crystallizations of bounded 4-manifolds, J. Knot Theory Ramifications 9(4) (2000) 443-458]) in order to significantly improve existing estimations for complexity of both 2-fold and three-fold simple branched coverings (see [O.M. Davydov, The complexity of 2-fold branched coverings of a 3-sphere, Acta Appl. Math. 75 (2003) 51-54] and [O.M. Davydov, Estimating complexity of 3-manifolds as of branched coverings, talk-abstract, Second Russian-German Geometry Meeting dedicated to 90-anniversary of A.D.Alexandrov, Saint-Petersburg, Russia, June 2002]) and 3-manifolds seen as Dehn surgery (see [G. Amendola, An algorithm producing a standard spine of a 3-manifold presented by surgery along a link, Rend. Circ. Mat. Palermo 51 (2002) 179-198]).     

Traces on the skein algebra of the torus

For a surface F, the Kauffman bracket skein module of F×[0,1], denoted K(F), admits a natural multiplication which makes it an algebra. When specialized at a complex number t, nonzero and not a root of unity, we have Kt(F), a vector space over C. In this paper, we will use the product-to-sum formula of Frohman and Gelca to show that the vector space Kt(T²) has five distinct traces. One trace, the Yang-Mills measure, is obtained by picking off the coefficient of the empty skein. The other four traces on Kt(T²) correspond to the four singular points of the moduli space of flat SU(2)-connections on the torus.     

Uniform 1-cochains and genuine laminations

We construct a pair of transverse genuine laminations on an atoroidal 3-manifold admitting a transversely orientable uniform 1-cochain. The laminations are induced by the uniform 1-cochain and they are the straightening of the coarse laminations defined by Calegari, by using minimal surface techniques. Moreover, when we collapse these laminations, we get a topological pseudo-Anosov flow, as defined by Mosher.     

Ganea and Whitehead definitions for the tangential Lusternik-Schnirelmann category of foliations

This work solves the problem of elaborating Ganea and Whitehead definitions for the tangential category of a foliated manifold. We develop these two notions in the category S-Top of stratified spaces, that are topological spaces X endowed with a partition F and compare them to a third invariant defined by using open sets. More precisely, these definitions apply to an element (X,F) of S-Top together with a class A of subsets of X; they are similar to invariants introduced by M. Clapp and D. Puppe.If (X,F)∈S-Top, we define a transverse subset as a subspace A of X such that the intersection S∩A is at most countable for any S∈F. Then we define the Whitehead and Ganea LS-categories of the stratified space by taking the infimum along the transverse subsets. When we have a closed manifold, endowed with a C¹-foliation, the three previous definitions, with A the class of transverse subsets, coincide with the tangential category and are homotopical invariants.     

Epimorphisms of knot groups onto free products

Let k be a knot in S³. There is an epimorphism from π₁(S³−k) onto a free product of two nontrivial cyclic groups sending a meridian to an element of length two iff k has property Q (Topology of Manifolds, Markham, Chicago, IL, 1970, pp. 195-199) that is if there is a closed surface F in S³ containing k, such that k is imprimitive in H₁(X) and in H₁(Y) where X and Y are the closures of the components of S³−F. We give answers to questions of Simon (1970) about properties Q, Q∗ and Q∗∗. Epimorphisms from knot groups onto torus knot groups are also studied and some results on property P and surgery are included.     

The self-amalgamation of high distance Heegaard splittings is always efficient

Let M be a compact orientable irreducible 3-manifold, F be an essential non-separating closed surface in M. We denote by η(F) the open regular neighborhood of F. If M−η(F) has a high distance Heegaard splitting, then M has a unique minimal Heegaard splitting up to isotopy.     

Orders and actions of branched coverings of hyperbolic links

Let M, M^′ be compact oriented 3-manifolds and L^′ a link in M^′ whose exterior has positive Gromov norm. We prove that the topological types of M and (M^′,L^′) determine the degree of a strongly cyclic covering branched over L^′. Moreover, if M^′ is a homology sphere then these topological types determine also the covering up to conjugacy.     

Embedding infinite cyclic covers of knot spaces into 3-space

We say a knot k in the 3-sphere S³ has PropertyIE if the infinite cyclic cover of the knot exterior embeds into S³. Clearly all fibred knots have Property IE.There are infinitely many non-fibred knots with Property IE and infinitely many non-fibred knots without property IE. Both kinds of examples are established here for the first time. Indeed we show that if a genus 1 non-fibred knot has Property IE, then its Alexander polynomial Δ_k(t) must be either 1 or 2t²−5t+2, and we give two infinite families of non-fibred genus 1 knots with Property IE and having Δ_k(t)=1 and 2t²−5t+2 respectively.Hence among genus 1 non-fibred knots, no alternating knot has Property IE, and there is only one knot with Property IE up to ten crossings.We also give an obstruction to embedding infinite cyclic covers of a compact 3-manifold into any compact 3-manifold.     

Topology of branched surfaces which admit expanding immersions

We investigate the topology of branched surfaces K which have the disjoint union of embedded circles as their branch sets SK, and which admit expanding immersions f with injective induced homomorphisms . If every connected component L of K?SK is orientable, then L is homeomorphic to a surface of genus ?1 with holes. In particular if there is a component homeomorphic to a 2-torus with holes, then K is the union of immersed tori. If every L is a 2-sphere with holes, under an additional assumption K is the union of immersed annuli.     

Incompressibility and least-area surfaces

We show that if F is a smooth, closed, orientable surface embedded in a closed, orientable 3-manifold M such that for each Riemannian metric g on M, F   is isotopic to a least-area surface F(g)$F(g)$, then F is incompressible.     

The hexatangle

We are interested in knowing what type of manifolds are obtained by doing Dehn surgery on closed pure 3-braids in S³. In particular, we want to determine when we get S³ by surgery on such a link. We consider links which are small closed pure 3-braids; these are the closure of 3-braids of the form , where σ₁, σ₂ are the generators of the 3-braid group and e₁, f₁, e are integers. We study Dehn surgeries on these links, and determine exactly which ones admit an integral surgery producing the 3-sphere. This is equivalent to determining the surgeries of some type on a certain six component link L that produce S³. The link L is strongly invertible and its exterior double branch covers a certain configuration of arcs and spheres, which we call the hexatangle. Our problem is equivalent to determine which fillings of the spheres by integral tangles produce the trivial knot, which is what we explicitly solve. This hexatangle is a generalization of the pentangle, which is studied in [C.McA. Gordon, J. Luecke, Non-integral toroidal Dehn surgeries, Comm. Anal. Geom. 12 (2004) 417-485].     

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 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.     

Toroidal and Klein bottle boundary slopes

Let M be a compact, connected, orientable, irreducible 3-manifold and T₀ an incompressible torus boundary component of M such that the pair (M,T₀) is not cabled. By a result of C. Gordon, if (S,∂S),(T,∂T)⊂(M,T₀) are incompressible punctured tori with boundary slopes at distance Δ=Δ(∂S,∂T), then Δ?8, and the cases where Δ=6,7,8 are very few and classified. We give a simplified proof of this result (or rather, of its reduction process), using an improved estimate for the maximum possible number of mutually parallel negative edges in the graphs of intersection of S and T. We also extend Gordon's result by allowing either S or T to be an essential Klein bottle.     

Rational structure on algebraic tangles and closed incompressible surfaces in the complements of algebraically alternating knots and links

Let F be an incompressible, meridionally incompressible and not boundary-parallel surface with boundary in the complement of an algebraic tangle (B,T). Then F separates the strings of T in B and the boundary slope of F is uniquely determined by (B,T) and hence we can define the slope of the algebraic tangle. In addition to the Conway's tangle sum, we define a natural product of two tangles. The slopes and binary operation on algebraic tangles lead to an algebraic structure which is isomorphic to the rational numbers.We introduce a new knot and link class, algebraically alternating knots and links, roughly speaking which are constructed from alternating knots and links by replacing some crossings with algebraic tangles. We give a necessary and sufficient condition for a closed surface to be incompressible and meridionally incompressible in the complement of an algebraically alternating knot or link K. In particular we show that if K is a knot, then the complement of K does not contain such a surface.     

A λ-lemma for foliations

We show that a C⁰ codimension one foliation with C¹ leaves F of a closed manifold is minimal if there are a foliation G transverse to F, and a diffeomorphism f preserving both foliations, such that every leaf of F intersects every leaf of G and f expands G. We use this result to study of Anosov actions on closed manifolds.     

3-manifolds which are orbit spaces of diffeomorphisms

In a very general setting, we show that a 3-manifold obtained as the orbit space of the basin of a topological attractor is either S²×S¹ or irreducible.We then study in more detail the topology of a class of 3-manifolds which are also orbit spaces and arise as invariants of gradient-like diffeomorphisms (in dimension 3). Up to a finite number of exceptions, which we explicitly describe, all these manifolds are Haken and, by changing the diffeomorphism by a finite power, all the Seifert components of the Jaco-Shalen-Johannson decomposition of these manifolds are made into product circle bundles.     

The Heegaard genera of surface sums

Let M be a compact orientable 3-manifold, and let F be a separating (resp. non-separating) incompressible surface in M which cuts M into two 3-manifolds M₁ and M₂ (resp. a manifold M₁). Then M is called the surface sum (resp. self surface sum) of M₁ and M₂ (resp. M₁) along F, denoted by M=M₁F_∪M₂ (resp. M=M₁F_∪). In this paper, we will study how g(M) is related to χ(F), g(M₁) and g(M₂) when both M₁ and M₂ have high distance Heegaard splittings.