Heiner Zieschang

Zur topologie gefaserter dreidimensionaler mannigfaltigkeiten

Zusammenfassung E s wird gezeigt, daβ zwei dreidimensionale Seifertsche Faserraume mit “hinreichend komplizierter Faserstruktur” genau dann durch einen fasertreuen Homoomorphismus ineinander uberfuhrt werden konnen, wenn ihre Fundamentalgruppen isomorph sind unter einem Isomorphismus, der die periphere Struktur erhalt. Hat die gewohnliche Faser einer Seifertschen Faserung unendliche Ordnung in der ersten Homologiegruppe, so gibt es eine “transversale” lokal-triviale Faserung des Raumes uber der S 1 mit einer Flache als Faser. Insbesondere trifft das fur alle berandeten orientierbaren Seifertschen Faserraume mit orientierbarer Zerlegungsflache zu.

On Heegaard decompositions of torus knot exteriors and related Seifert fibre spaces

The Heegaard decompositions of genus 2 oftorus knot exteriors are classified with respect to homeomorphisms. It turns out that in general there are three different classes which are also the isotopy classes. A similar result is obtained for Seifert fibre spaces, having a disk as base and two exceptional fibres such as the exteriors of torus knots have or having a sphere as base and three exceptional fibres.

Realization of primitive branched coverings over closed surfaces following the hurwitz approach

AbstractLet V be a closed surface, H⊑π1(V) a subgroup of finite index l and D=[A1,...,Am] a collection of partitions of a given number d≥2 with positive defect v(D). When does there exist a connected branched covering f:W→V of order d with branch data D and f∶W→V It has been shown by geometric arguments [4] that, for l=1 and a surface V different from the sphere and the projective plane, the corresponding branched covering exists (the data D is realizable) if and only if the data D fulfills the Hurwitz congruence v(D)э0 mod 2. In the case l>1, the corresponding branched covering exists if and only if v(D)э0 mod 2, the number d/l is an integer, and each partition Ai∈D splits into the union of l partitions of the number d/l. Here we give a purely algebraic proof of this result following the approach of Hurwitz [11].The realization problem for the projective plane and l=1 has been solved in [7,8]. The case of the sphere is treated in [1, 2, 12, 7].

The cohomology ring of a class of Seifert manifolds

Abstract The cohomology groups of the Seifert manifolds are well known. In this article a method is given to compute the cup products in the cohomology ring of any orientable Seifert manifold whose associated orbit surface is S 2 , and for any coefficients. In particular the Z /2 cohomology ring is completely determined. This is applied to determine the existence of degree 1 maps from the Seifert manifold to R P 3 , and to the Lusternik–Schnirelmann category.

Realization of Primitive Branched Coverings Over Closed Surfaces

Let V be a closed surface, H ⊆ π1(V ) a subgroup of finite index and D = [A1, . . . , Am] a collection of partitions of a given number d ≥ 2 with positive defect v(D). When does there exist a connected branched covering f : W → V of order d with branch data D and f#(π1(W )) = H? We show that, for a surface V different from the sphere and the projective plane and = 1, the corresponding branched covering exists (the data D is realizable) if and only if the data D fulfills the Hurwitz congruence v(D) ≡ 0 mod 2. In the case > 1, the corresponding branched covering exists if and only if v(D) ≡ 0 mod 2, the number d/ is an integer, and each partition Ai ∈ D splits into the union of partitions of the number d/ . The realization problem for the projective plane and = 1 has been solved in (Edmonds-Kulkarni-Stong, 1984). The case of the sphere is treated in (BersteinEdmonds, 1979; Berstein-Edmonds, 1984; Husemoller, 1962; Edmonds-KulkarniStong, 1984). AMS: Primary: 55M20, Secondary: 57M12, 20F99

Computer Calculation of the Degree of Maps into the Poincaré Homology Sphere

Let M and P be Seifert 3-manifolds. Is there a degree one map f : M → P? The problem was completely solved by Hayat Legrand, Wang, and Zieschang for all casesexcept when P is the Poincaré homology sphere. We investigate the remaining case by elaborating and implementing a computer algorithm that calculates the degree. As a result, we get an explicit experimental expression for the degree through numerical invariants of the induced homomorphism f# : π1(M) → π1(P).

On the cohomology of Seifert and graph manifolds

Abstract In this survey the cohomology rings H ∗ (M 3 ; Z 2 ) of orientable Seifert and graph manifolds are described and some proofs are sketched.

Any 3-manifold 1-dominates at most finitely many 3-manifolds of S^3-geometry

Any 3-manifold 1-dominates at most finitely many 3-manifolds supporting S3 geometry.

Primitive elements in the free product of two finite cyclic groups

For the free product 〈s |sa=1〉*〈t|tb=1〉 we give a precise description of the wordsu ins,t which are primitive (i.e. there exists an elementv such thatu,v generate the whole group).

Some quadratic equations in the free group of rank 2

For a given quadratic equation with any number of unknowns in any free group F , with right-hand side an arbitrary element of F , an algorithm for solving the problem of the existence of a solution was given by Culler [8] using a surface method and generalizing a result of Wicks [46]. Based on different techniques, the problem has been studied by the authors [11; 12] for parametric families of quadratic equations arising from continuous maps between closed surfaces, with certain conjugation factors as the parameters running through the group F . In particular, for a oneparameter family of quadratic equations in the free group F2 of rank 2, corresponding to maps of absolute degree 2 between closed surfaces of Euler characteristic 0, the problem of the existence of faithful solutions has been solved in terms of the value of the self-intersection index W F2! ZaF2c on the conjugation parameter. The present paper investigates the existence of faithful, or non-faithful, solutions of similar families of quadratic equations corresponding to maps of absolute degree 0. The existence results are proved by constructing solutions. The non-existence results are based on studying two equations in Zac and in its quotient Q, respectively, which are derived from the original equation and are easier to work with, where is the fundamental group of the target surface, and Q is the quotient of the abelian group Za nf1gc by the system of relations g g 1 , g2 nf1g. Unknown variables of the first and second derived equations belong to , Zac , Q, while the parameters of these equations are the projections of the conjugation parameter to and Q, respectively. In terms of these projections, sufficient conditions for the existence, or non-existence, of solutions of the quadratic equations in F2 are obtained.