Friedrich Hegenbarth

On manifold spines and cyclic presentations of groups

This is a survey of results and open problems on compact 3-manifolds which admit spines corresponding to cyclic presentations of groups. We also discuss questions concerning spines of knot manifolds and regular neighborhoods of homotopically PL embedded compacta in 3-manifolds. 1. Spines of 3-manifolds. Let G = 〈x1, x2, . . . , xn : g1, g2, . . . , gm〉 be a finite group presentation with n generators and m relators, n ≥ m. We can associate to G a canonical 2-complex KG, with one vertex v, such that Π1(KG) is presented by G. Its 1-skeleton K (1) G is a bouquet of n circles with a fixed orientation, also denoted 1991 Mathematics Subject Classification: Primary 57M05; Secondary 57M12. Work performed under the auspices of the G.N.S.A.G.A. of the C.N.R. (National Research Council) of Italy and partially supported by the Ministero per la Ricerca Scientifica e Tecnologica of Italy within the projects Geometria Reale e Complessa and Topologia and by the Ministry of science and Technology of the Republic of Slovenia grant No. J1-7039-0101-95. The paper is in final form and no version of it will be published elsewhere.

On 4-manifolds with free fundamental group

We study the homotopy type of closed 4-manifolds (or oriented Poincare spaces) with free fundamental group. This gives a partial solution to problem N. 4.53 of [10]. Then we extend the Whitehead-Novikov-Wall theorem for this class of manifolds by using surgery techniques. 1991 Mathematics Subject Classification: 57N65, 57R67; 57Q10, 57R80.

Four-Manifolds With Surface Fundamental Groups

We study the homotopy type of closed connected topological 4manifolds whose fundamental group is that of an aspherical surface F . Then we use surgery theory to show that these manifolds are s-cobordant to connected sums of simply-connected manifolds with an S2-bundle over F .

On cyclic bra nched coverings of torus knots

We prove that then-fold cyclic coverings of the 3-sphere branched over the torus knotsK(p,q), p>q≥2 (i.e. the Brieskorn manifolds in the sense of [12]) admit spines corresponding to cyclic presentations of groups ifp≡1 (modq). These presentations include as a very particular case the Sieradski groups, first introduced in [14] and successively obtained from geometric constructions in [4], [9], and [15]. So our main theorem answers in affirmative to an open question suggested by the referee in [14]. Then we discuss a question concerning cyclic presentations of groups and Alexander polynomials of knots.

On the determination of PL-manifolds by handles of lower dimension☆

Abstract We extend the Montesinos theorem about handle presentations of PL 4-manifolds to dimension n . For this we consider closed PL ( n +1)-manifolds of the same homotopy type as X = # p ( S 1 X S n ). Then we extend PL homeomorphisms of X over the solid handlebody.

On the homotopy classification of 4-manifolds having the fundamental group of an aspherical 4-manifold

In this paper we shall study the homotopy type of closed connected oriented topological 4-manifolds M with fundamental group isomorphic to Πi( • <2 be the classifying map of the universal covering. For this we shall prove the following result (see Section 3).

On minimal Poincaré 4-complexes

We consider 2 types of minimal Poincare 4-complexes. One is defined with respect to the degree 1-map order. This idea was already present in our previous papers, and more systematically studied later by Hillman. The second type of minimal Poincare 4-complexes was introduced by Hambleton, Kreck, and Teichner. It is not based on an order relation. In the present paper we study existence and uniqueness questions.

Surgery on 3-manifolds with \(\mathbb{S}\)1-actions

AbstractWe study the topological structure of all 3-manifolds obtained by surgery along principal fibers of a closed orientable

Manifolds with Highly Connected Universal Covers

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