Narrow Results

We classify smooth projective algebraic curves C of genus g such that the variety of special linear systems has dimension g- 7. We first prove that if has dimension g-7≥0 then C is either trigonal, tetragonal, a double covering of a curve of genus 2 or a smooth plane sextic. This result establishes the next extension of dimension theorems of H. Martens and D. Mumford on the variety of special linear systems with the fullest possible generality. We then proceed to show that, under the assumption g≥11, has dimension g- 7 if and only if C is either a trigonal curve or a double covering of a curve of genus 2.

For any n>1, we construct examples of branched Galois coverings M→ℙⁿ where M is one of (ℙ¹)ⁿ, and , where is the 1-ball. In terms of orbifolds, this amounts to giving examples of orbifolds over ℙⁿ uniformized by M. We also discuss the related "orbifold braid groups".

We study ramified covers of the projective plane ℙ². Given a smooth surface S in ℙ^N and a generic enough projection ℙ^N → ℙ², we get a cover π: S → ℙ², which is ramified over a plane curve B. The curve B is usually singular, but is classically known to have only cusps and nodes as singularities for a generic projection. The main question that arises is with respect to the geometry of branch curves; i.e. how can one distinguish a branch curve from a non-branch curve with the same numerical invariants? For example, a plane sextic with six cusps is known to be a branch curve of a generic projection iff its six cusps lie on a conic curve, i.e. form a special 0-cycle on the plane. The classical work of Beniamino Segre gives a complete answer to the second question in the case when S is a smooth surface in ℙ³. We give an interpretation of the work of Segre in terms of relation between Picard and Chow groups of 0-cycles on a singular plane curve B. In addition, the appendix written by E. Shustin shows the existence of new Zariski pairs.

The paper relates the 4-fold symmetric quandle homotopy (cocycle) invariants to topological objects. We show that the 4-fold symmetric quandle homotopy invariants are at least as powerful as the Dijkgraaf–Witten invariants. As an application, for an odd prime p, we show that the quandle cocycle invariant of a link in S³ constructed by the Mochizuki 3-cocycle is equivalent to the Dijkgraaf–Witten invariant with respect to ℤ/pℤ of the double covering of S³ branched along the link. We also reconstruct the Chern–Simons invariant of closed 3-manifolds as a quandle cocycle invariant via the extended Bloch group, in analogy to [A. Inoue and Y. Kabaya, Quandle homology and complex volume, preprint(2010), arXiv:math/1012.2923].

We give a brief survey of the so-called Fenchel's problem for the projective plane, that is the problem of existence of finite Galois coverings of the complex projective plane branched along a given divisor and prove the following result: Let p, q be two integers greater than 1 and C be an irreducible plane curve. If there is a surjection of the fundamental group of the complement of C into a free product of cyclic groups of orders p and q, then there is a finite Galois covering of the projective plane branched along C with any given branching index.

Extending and reproving a recent result of D. Krebes, we give obstructions to the embedding of a tangle in a link.

We introduce a family of cyclic presentations of groups depending on a finite set of integers. This family contains many classes of cyclic presentations of groups, previously considered by several authors. We prove that, under certain conditions on the parameters, the groups defined by our presentations cannot be fundamental groups of closed connected hyperbolic 3–dimensional orbifolds (in particular, manifolds) of finite volume. We also study the split extensions and the natural HNN extensions of these groups, and determine conditions on the parameters for which they are groups of 3–orbifolds and high–dimensional knots, respectively.

We consider a family of words in a free group of rank n which determine 3-manifolds ℳ_n(p,q). We prove that the fundamental groups of ℳ_n(p,q) are cyclically presented, and that ℳ_n(p,q) is the n-fold cyclic covering of the 3-sphere branched over the torus knots T(p,q) if p is odd and q≡±2(mod p). We also obtain an explicit Dunwoody parameters for the torus knots T(p,q) for odd p and q≡±2(mod p).

We give explicit palindrome presentations of the groups of rational knots, i.e. presentations with relators which read the same forwards and backwards. This answers a question posed by Hilden, Tejada and Toro in 2002. Using such presentations we obtain simple alternative proofs of some classical results concerning the Alexander polynomial of all rational knots and the character variety of certain rational knots. Finally, we derive a new recursive description of the SL(2, ℂ) character variety of twist knots.

We show that for any n ≥ 4 there exists an equivalence functor from the category of n-fold connected simple coverings of B³ × [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, and the cobordism category of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S³ branched over links, which provides a complete solution to the long-standing Fox–Montesinos covering moves problem. This last result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of the equivalence theorem to possibly non-simple coverings of S³ branched over embedded graphs. Then, we factor the functor above as , where is an equivalence functor to a universal braided category freely generated by a Hopf algebra object H. In this way, we get a complete algebraic description of the category . From this we derive an analogous description of the category of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.

The main result of this paper is the construction of two Hyperbolic manifolds, $M_1$ and $M_2$, with several remarkable properties:

• ⁽¹⁾Every closed orientable $3$-manifold is homeomorphic to the quotient space of the action of a group of order $16$ on some covering space of $M_1$ or $M_2$.
• ⁽²⁾$M_1$ and $M_2$ are tesselated by 16 dodecahedra such that the pentagonal faces of the dodecahedra fit together in a certain way.
• ⁽³⁾There are 12 closed non-orientable hyperbolic surfaces of Euler characteristic $-2$ each of which is tesselated by regular right angled pentagons and embedded in $M_1$ or $M_2$. The union of the pentagonal faces of the tesselating dodecahedra equals the union of the 12 images of the embedded surfaces of Euler characteristic $-2$.

Let $t_{\alpha ,\beta }\subset S^2\times S^1$ be an ordinary fiber of a Seifert fibering of $S^2\times S^1$ with two exceptional fibers of order $\alpha$. We show that any Seifert manifold with Euler number zero is a branched covering of $S^2\times S^1$ with branching $t_{\alpha ,\beta }$ if $\alpha \ge 3$. We compute the Seifert invariants of the Abelian covers of $S^2\times S^1$ branched along a $t_{\alpha ,\beta }$. We also show that $t_{2,1}$, a non-trivial torus knot in $S^2\times S^1$, is not universal.

The $d$-fold ($d\ge 3$) branched coverings on a disk give an infinite family of nongeometric embeddings of braid groups into mapping class groups. We, in this paper, give new explicit expressions of these braid group representations into automorphism groups of free groups in terms of the actions on the generators of free groups. We also give a systematic way of constructing and expressing these braid group representations in terms of a new gadget, called covering groupoid. We prove that each generator ${\tilde{\beta }}_i$ of braid group inside mapping class group induced by $d$-fold covering is the product of $d-1$ Dehn twists on the surface.

In this paper, we study closed orientable Euclidean manifolds which are also known as flat three-dimensional manifolds or just Euclidean 3-forms. Up to homeomorphism, there are six of them. The first one is the three-dimensional torus. In 1972, Fox showed that the 3-torus is not a double branched covering of the 3-sphere. So, it is not a hyperelliptic manifold. In this paper, we show that all the remaining Euclidean 3-forms are hyperelliptic manifolds.

Closed hyperbolic 3-manifolds obtained by Dehn surgeries on the Whitehead link yield interesting examples of manifolds of small volume. In the present paper these manifolds are described as 2-fold coverings of the 3-sphere branched over 3-bridge links. As a corollary, maximally symmetric -manifolds of small volume are obtained.

Kim and Kostrikin constructed in [8, 9] a tessellation on the boundary of a polyhedral 3-cell consisting of 8n pentagons (n ≥ 1). The tessellation produces a family of closed connected orientable 3-manifolds (denoted by M₁(n) in the quoted papers) with spines corresponding to certain presentations of their fundamental groups. We investigate the topological and algebraic properties of such groups together with their derived quotients and split extensions, and completely classify the considered manifolds.

Dunwoody manifolds are an interesting class of closed connected orientable 3-manifolds, which are defined by means of Heegaard diagrams having a rotational symmetry. They are proved to be cyclic coverings of lens spaces (possibly 𝕊³) branched over (1,1)-knots. Here we study the Dunwoody manifolds which are cyclic coverings of the 3-sphere branched over two specified families of Montesinos knots. Then we determine the Dunwoody parameters for such knots and the isometry groups for the considered manifolds in the hyperbolic case. A list of volumes for some hyperbolic Dunwoody manifolds completes the paper.