Alberto Cavicchioli

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.

A survey on snarks and new results: Products, reducibility and a computer search

In this paper we survey recent results and problems of both theoretical and algorithmic character on the construction of snarksnon-trivial cubic graphs of class two, of cyclic edge-connectivity at least 4 and with girth ≥ 5. We next study the process, also considered by Cameron, Chetwynd, Watkins, Isaacs, Nedela, and Skoviera, of splitting a snark into smaller snarks which compose it. This motivates an attempt to classify snarks by recognizing irreducible and prime snarks and proving that all snarks can be constructed from them. As a consequence of these splitting operations, it follows that any snark (other than the Petersen graph) of order ≤ 26 can be built as either a dot product or a square product of two smaller snarks. Using a new computer algorithm we have confirmed the computations of Brinkmann and Steffen on the classification of all snarks of order less than 30. Our results recover the well-known classification of snarks of order not exceeding 22. Finally, we prove that any snark G of order ≤ 26 is almost Hamiltonian, in the sense that G has at least one vertex v for which G \ v is Hamiltonian.

TOPOLOGICAL PROPERTIES OF CYCLICALLY PRESENTED GROUPS

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.

Special Classes of Snarks

We report the most relevant results on the classification, up to isomorphism, of nontrivial simple uncolorable (i.e., the chromatic index equals 4) cubic graphs, called snarks in the literature. Then we study many classes of snarks satisfying certain additional conditions, and investigate the relationships among them. Finally, we discuss connections between the snark family and some significant conjectures of graph theory, and list some problems and open questions which arise naturally in this research.

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.

Neuwirth manifolds and colourings of graphs

SummaryThe paper uses the fact that PL manifolds may be studied through graphs with coloured edges. The representation is given by taking the l-skeleton of the cellular subdivision dual to a suitable triangulation (minimal with respect to the vertices) of a manifold. Here we describe a very simple construction to obtain a 4-coloured graph representing a closed orientable 3-manifold from its Heegaard diagram. This construction allows us to completely classify a countable class of closed orientable 3-manifoldsMn,n ⩾ 3, introduced by L. Neuwirth in Proc. Camb. Phil. Soc.64 (1968), 603–613. Indeed, we show thatMn is homeomorphic to the Seifert fibered space

Four-Manifolds With Surface Fundamental Groups

\sum _n = (0 \circ 0 / - 1 \underbrace {(2,1) (2,1) ... (2,1)).}_{n times}

Imbeddings of polyhedra in 3-manifolds

It is also proved thatMn is the unique closed 3-manifold having the canonical 2-complex associated to the standard presentation of the Fibonacci groupF(n − 1, n) as its spine.