We explore the constraints imposed by Poincaré duality on the resonance varieties of a graded algebra. For a three-dimensional Poincaré duality algebra A, we obtain a fairly precise geometric description of the resonance varieties ${\cal R}^i_k(A)$.

Combings of compact, oriented, 3-dimensional manifolds $M$ are homotopy classes of nowhere vanishing vector fields. The Euler class of the normal bundle is an invariant of the combing, and it only depends on the underlying $\text{Spi}{{\text{n}}^{\text{c}}}$-structure. A combing is called torsion if this Euler class is a torsion element of ${{H}^{2}}(M;\,\mathbb{Z})$. Gompf introduced a $\mathbb{Q}$-valued invariant ${{\theta }_{G}}$ of torsion combings on closed 3-manifolds, and he showed that ${{\theta }_{G}}$ distinguishes all torsion combings with the same $\text{Spi}{{\text{n}}^{\text{c}}}$-structure. We give an alternative definition for ${{\theta }_{G}}$ and we express its variation as a linking number. We define a similar invariant ${{p}_{1}}$ of combings for manifolds bounded by ${{S}^{2}}$. We relate ${{p}_{1}}$ to the $\Theta$-invariant, which is the simplest configuration space integral invariant of rational homology 3-balls, by the formula $\Theta \,=\,\frac{1}{4}{{p}_{1}}\,+\,6\text{ }\!\!\lambda\!\!\text{ }\left( {\hat{M}} \right)$, where $\text{ }\!\!\lambda\!\!\text{ }$ is the Casson-Walker invariant. The article also includes a self-contained presentation of combings for 3-manifolds.

A subset $W$ of a closed manifold $M$ is $K$-contractible, where $K$ is a torus or Klein bottle if the inclusion $W\,\to \,M$ factors homotopically through a map to $K$. The image of ${{\pi }_{1}}\left( W \right)$ (for any base point) is a subgroup of ${{\pi }_{1}}\left( M \right)$ that is isomorphic to a subgroup of a quotient group of ${{\pi }_{1}}\left( K \right)$. Subsets of $M$ with this latter property are called ${{\mathcal{G}}_{K}}$-contractible. We obtain a list of the closed 3-manifolds that can be covered by two open ${{\mathcal{G}}_{K}}$-contractible subsets. This is applied to obtain a list of the possible closed prime 3-manifolds that can be covered by two open $K$-contractible subsets.

This paper contains some applications of the description of knot diagrams by genus, and Gabai’s methods of disk decomposition. We show that there exists no genus one knot of canonical genus 2, and that canonical genus 2 fiber surfaces realize almost every Alexander polynomial only finitely many times (partially confirming a conjecture of Neuwirth).

This paper concerns the problem of existence of taut foliations among 3-manifolds. From the work of David Gabai we know that a closed 3-manifold with non-trivial second homology group admits a taut foliation. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite different if they are integral or rational but non-integral homology 3-spheres. Concerning integral homology 3-spheres, we can see that all but the 3-sphere and the Poincaré 3-sphere admit a taut foliation. Concerning non-integral homology 3-spheres, we prove there are infinitely many that admit a taut foliation, and infinitely many without a taut foliation. Moreover, we show that the geometries do not determine the existence of taut foliations on non-integral Seifert fibered homology 3-spheres.

For the mapping class group of 3-manifold with respect to a Heegaard splitting, a simplicial complex is constructed such that its group of automorphisms is identified with the mapping class group.

For a hyperbolic 3-manifold $M$ with a torus boundary component, all but finitely many Dehn fillings yield hyperbolic 3-manifolds. In this paper, we will focus on the situation where $M$ has two exceptional Dehn fillings: an annular filling and a toroidal filling. For such a situation, Gordon gave an upper bound of 5 for the distance between such slopes. Furthermore, the distance 4 is realized only by two specific manifolds, and 5 is realized by a single manifold. These manifolds all have a union of two tori as their boundaries. Also, there is a manifold with three tori as its boundary which realizes the distance 3. We show that if the distance is 3 then the boundary of the manifold consists of at most three tori.

Let $B_n$ denote the classical braid group on $n$ strands and let the mixed braid group$B_{m,n}$ be the subgroup of $B_{m+n}$ comprising braids for which the first $m$ strands form the identity braid. Let $B_{m,\infty}=\bigcup_nB_{m,n}$. We describe explicit algebraic moves on $B_{m,\infty}$ such that equivalence classes under these moves classify oriented links up to isotopy in a link complement or in a closed, connected, oriented three-manifold. The moves depend on a fixed link representing the manifold in $S^3$. More precisely, for link complements the moves are the two familiar moves of the classical Markov equivalence together with ‘twisted’ conjugation by certain loops $a_i$. This means premultiplication by $a_i^{-1}$ and postmultiplication by a ‘combed’ version of $a_i$. For closed three-manifolds there is an additional set of ‘combed’ band moves that correspond to sliding moves over the surgery link. The main tool in the proofs is the one-move Markov theorem using $L$-moves (adding in-box crossings). The resulting algebraic classification is a direct extension of the classical Markov theorem that classifies links in $S^3$ up to isotopy, and potentially leads to powerful new link invariants, which have been explored in special cases by the first author. It also provides a controlled range of isotopy moves, useful for studying skein modules of three-manifolds.

Let $K$ be a knot in ${{S}^{3}}$ . This paper is devoted to Dehn surgeries which create 3-manifolds containing a closed non-orientable surface $\hat{S}$ . We look at the slope $p/q$ of the surgery, the Euler characteristic $\mathcal{X}(\hat{S})$ of the surface and the intersection number $s$ between $\hat{S}$ and the core of the Dehn surgery. We prove that if $\mathcal{X}(\hat{S})\,\ge \,15\,-3q$, then $s\,=\,1$. Furthermore, if $s\,=\,1$ then $q\,\le \,4\,-\,3\,\mathcal{X}(\hat{S})$ or $K$ is cabled and $q\,\le \,8\,-5\mathcal{X}(\hat{S})$ . As consequence, if $K$ is hyperbolic and $\mathcal{X}(\hat{S})\,=\,-1$ , then $q\,\le \,7$.

The infimal Heegaard gradient of a 3-manifold was defined and studied by Marc Lackenby in an approach towards proving the well-known virtually Haken conjecture. As instructive examples, Seifert fibered 3-manifolds are considered in this paper. The author shows that a compact orientable Seifert fibered 3-manifold has zero infimal Heegaard gradient if and only if it virtually fibers over either the circle or a surface other than the 2-sphere or, equivalently, if it has infinite fundamental group.

If a hyperbolic link has a prime alternating diagram $D$, then we show that the link complement's volume can be estimated directly from $D$. We define a very elementary invariant of the diagram $D$, its twist number $t(D)$, and show that the volume lies between $v_3(t(D) - 2)/2$ and $v_3(10t(D) - 10)$, where $v_3$ is the volume of a regular hyperbolic ideal 3-simplex. As a consequence, the set of all hyperbolic alternating and augmented alternating link complements is a closed subset of the space of all complete finite-volume hyperbolic 3-manifolds, in the geometric topology.

Using the Fiedler–Polyak–Viro Gauß diagram formulae we study the Vassiliev invariants of degree 2 and 3 on almost positive knots. As a consequence we show that the number of almost positive knots of a given genus or unknotting number grows polynomially in the crossing number, and also recover and extend, inter alia to their untwisted Whitehead doubles, previous results on the polynomials and signatures of such knots. In particular, we prove that there are no achiral almost positive knots and classify all almost positive diagrams of the unknot. We give an application to contact geometry (Legendrian knots) and property P.

Let N be a finitely generated normal subgroup of a finitely generated group G. We show that if the trivial subgroup is tame in the factor group G/N, then N is that in G. We also give a short new proof of the fact that quasiconvex subgroups of negatively curved groups are tame. The proof utilizes the concept of the geodesic core of the subgroup and is related to the Dehn algorithm.

We show that diagrammatically reducible two-complexes are characterized by the property: every finity subconmplex of the universal cover collapses to a one-complex. We use this to show that a compact orientable three-manifold with nonempty boundary is Haken if and only if it has a diagrammatically reducible spine. We also formulate an nanlogue of diagrammatic reducibility for higher dimensional complexes. Like Haken three-manifolds, we observe that if n ≥ 4 and M is compact connected n-dimensional manifold with a traingulation, or a spine, satisfying this property, then the interior of the universal cover of M is homeomorphic to Euclidean n-space.

Any knot in a 3-dimensional homology sphere is homotopic to a knot with trivial Alexander polynomial.

We outline the classification, up to isometry, of all tetrahedra in hyperbolic space with one or more vertices truncated, for which the dihedral angles along the edges formed by the truncations are all π/2, and those remaining are all submultiples of π. We show how to find the volumes of these polyhedra, and find presentations and small generating sets for the orientation-preserving subgroups of their reflection groups.

For particular families of these groups, we find low index torsion free subgroups, and construct associated manifolds and manifolds with boundary. In particular, for each g ≥ 2, we find a sequence of hyperbolic manifolds with totally geodesic boundary of genus g, which we conjecture to be of least volume among such manifolds.

We classify the compact 3-manifolds whose boundary is a union of 2-spheres, and which embed in T ×I ×I, where T is a triod and I the unit interval. This class is described explicitly as the set of punctured handlebodies. We also show that any 3-manifold in T × I × I embeds in a punctured handlebody.

We show that every Turaev-Viro invariant for 3-manifolds is a sum of three new invariants and discuss their properties. We also find a solution of a conjecture of L. H. Kauffman and S. Lins. Tables of the invariants for closed orientable 3-manifolds of complexity ≤ 3 are presented at the end of the paper.

This paper defines three simplicial approximation properties for maps of 2-cells and 2-spheres into spaces, each providing homotopical tameness conditions on the approximating images. These are the general position properties used in the two main results. The first shows that a resolvable generalized 3-manifold is a genuine 3- manifold if and only if it has the weakest of these approximation properties as well as a mild 3-dimensional disjoint disks condition known as the Light Map Separation Property. The second shows a resolvable generalized 3-manifold to be a 3-manifold if and only if it satisfies the strongest of these approximation properties.

We give a relation between the homology groups H1() and H1 (M) for a branched cyclic cover → M of arbitrary closed, oriented 3-manifolds which generalizes a classical result of Plans on covers of S3 branched over a knot and provides other quantitative information as well. We include a general "free calculus" procedure for computing homology groups of branched covers and reinterpret the results in this computational setting.