We consider some families of one relator groups arising as fundamental groups of 3-dimensional manifolds, and calculate their character varieties in SL(2, C). Then we give simple geometrical descriptions of such varieties, and determine the number of their irreducible components. Our paper relates to the work of Baker-Petersen, Qazaqzeh and Morales-Marcén on the character variety of certain classes of one relator groups, but we use different methods based on the concept of palindrome presentations of given groups.

The Browder-Livesay invariants provide obstructions to the realization of elements of Wall groups by normal maps of closed manifolds. A generalization of the iterated Browder-Livesay invariants is proposed and properties of the invariants obtained are described. The generalized definition makes it possible to investigate the relationship between a normal map and its restriction to a submanifold and clarifies the relationship between the Browder-Livesay invariants and the Browder-Quinn groups of obstructions to surgery on filtered manifolds. Several theorems describing a relationship between a normal map and its restriction to a submanifold are proved.

A closed 3-manifold M is said to be hyperelliptic if it has an involution τ such that the quotient space of M by the action of τ is homeomorphic to the standard 3-sphere. We show that the hyperbolic football manifolds of Emil Molnár [12] are hyperelliptic. Then we determine the isometry groups of such manifolds. Another consequence is that the unique hyperbolic dodecahedral and icosahedral 3-space forms with first homology group ℤ35 (constructed by I. Prok in [16], on the basis of a principal algorithm due to Emil Molnár [13], and by Richardson and Rubinstein in [18]) are also hyperelliptic.

The problem of classifying, up to isometry, the orientable 3-manifolds that arise by identifying the faces of a Platonic solid was completely solved in a nice paper of Everitt [B. Everitt, 3-manifolds from Platonic solids, Topology Appl. 138 (2004) 253–263]. His work completes the classification begun by Best [L.A. Best, On torsion-free discrete subgroups of PSL2(C) with compact orbit space, Canad. J. Math. 23 (1971) 451–460], Lorimer [P.J. Lorimer, Four dodecahedral spaces, Pacific J. Math. 156 (2) (1992) 329–335], Prok [I. Prok, Classification of dodecahedral space forms, Beiträge Algebra Geom. 39 (2) (1998) 497–515], and Richardson and Rubinstein [J. Richardson, J.H. Rubinstein, Hyperbolic manifolds from a regular polyhedron, Preprint]. In this paper we investigate the topology of closed orientable 3-manifolds from Platonic solids. Here we completely recognize those manifolds in the spherical and Euclidean cases, and state topological properties for many of them in the hyperbolic case. The proofs of the latter will appear in a forthcoming paper.

We describe a simple algorithm to obtain a catalogue of 3-bridge links by computer, depending upon 6-tuples of positive integers. This permits us to represent the genus two 3-manifolds by standardly constructed graphs with colored edges. Finally, we prove some results about the topological structure of these manifolds and extend the combinatorial representation to n-bridge links.

OBJECTIVE:To compare the effectiveness of a high-specification foam mattress (control) with a high-tech (Duo2, Hill Rom) alternating/continuous low-pressure mattress (treatment) in the prevention of pressure ulceration. The study also evaluated if there is a difference in performance between the two working modalities (alternating and continuous low pressure) of the high-tech mattress in a comparable sample of patients.METHOD:Thirty-three patients were observed for two weeks in the control group. In the treatment group, 86 patients were randomised to receive alternating low pressure and 84 continuous low pressure. Incidence of pressure ulcers in both arms was recorded. Student's t-test was used to compare all Braden scores, and the chi-square test and Fisher's exact test to evaluate differences between groups.RESULTS:There was a high difference in the number of new pressure ulcers in the control group when compared with the treatment group. There was no difference in performance between the alternating and continuous low-pressure modes. However, the sample size is too small to prove or disprove a statistically significant difference between the two modalities.CONCLUSION:The high-tech mattress was markedly more effective than the high-specification foam mattress in preventing the onset of pressure ulcers. Initial data suggest that the use of alternating or continuous low pressure made little or no difference to the results.

We consider orientable closed connected 3‐manifolds obtained by Dehn surgeries with rational coefficients along the components of certain periodic links. These manifolds extend many classes of (hyperbolic) manifolds considered by several authors (see the references). We find geometric presentations of the fundamental group of such manifolds, and study some covering properties of them. Then we obtain results on their geometric structures in many cases. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

A kinetics study on PbGeO3 solid–solid phase transition was realised by means of differential scanning calorimetry and time-resolved X-ray powder diffraction. Isothermal and non-isothermal Johnson–Mehl–Avrami equations were applied to obtain both the activation energy Ea and the Avrami coefficient n. The latter parameter has been related to morphological evidences collected by scanning electron microscopy. The limits of the applied theory, i.e., the Arrhenian behaviour of the growth rate and a steady-state nucleation rate, are finally discussed in terms of recent theoretical developments.

In this paper we consider various types of relative groups which naturally arise in surgery theory, and describe algebraic properties of them. Then we apply the obtained results to investigate the splitting obstruction groups LS* and the surgery obstruction groups LP* for a manifold pair. Finally, we introduce the lower LS*- and LP*-groups, and describe connections between them and the corresponding lower L-*-groups and surgery exact sequence.

where the subscripts are reduced modulo n, and r ≥ 2, n ≥ 2. We also denote in short the above presentation by F (r, n, s, p) = Gn(x1x1+p · · ·x1+p(r−1)x 1+s) according to notation in [13] and [16]. This family contains many classes of cyclic presentations of groups, previously considered by several authors. The groups F (2, n, 2, 1) are the Fibonacci groups F (2, n) = Gn(x1x2x 3 ) introduced by Conway in [9] (see also [10]). The groups F (r, n, r, 1) = Gn(x1x2 · · ·xrx 1+r) were introduced by Johnson, Wamsley and Wright in [17] (and denoted by F (r, n), r ≥ 2, n ≥ 3) as a natural generalization of the Fibonacci groups F (2, n) (and in fact they are called with the same name in the current literature). The groups F (2, n, 1, 2) = Gn(x1x3x 2 ) are the Sieradski groups introduced in [22] (and denoted by S(n)) (see also [5] for some generalizations of them). The groups F (r, n, r+k−1, 1) = Gn(x1x2 · · ·xrx r+k) were defined and algebraically studied by Campbell and Robertson in [1] (and denoted by F (r, n, k)) for any r ≥ 2, n ≥ 3, and k ≥ 1. Obviously, they represent further generalizations of the Fibonacci groups F (r, n). The groups F (2, n, 1, p) = Gn(x1x1+px 2 ) are the Gilbert–Howie groups defined in [11], and denoted by H(n, p). A natural generalization of them, that is, F (2, n, s, p) = Gn(x1x1+px 1+s), was considered in [6], and denoted by Gn(p, s). Obviously, we have Gn(2, 1) = S(n) and Gn(1, 2) = F (2, n). In this paper we study algebraic systems, or briefly algebras, which are groups with an additional unary operation satisfying certain laws. These laws are directly suggested by the relators of the cyclically presented groups F (r, n, s, p). More precisely, in addition to the group laws (we use notation x · y, x−1, and e for the product, inverse, and unit element, respectively) there is an unary operation φ (written as a right–hand operation) which satisfies the following properties:

We construct 4k-dimensional generalized manifolds, k > 1, which have no resolutions. The construction proceeds as in a paper of Bryant, Ferry, Mio and Weinberger (see [1]) but does not use their controlled (epsilon, delta)-surgery sequence. The controlled surgery sequence is believed to be true. Recently, Pedersen, Quinn and Ranicki have given a proof of this sequence in the case of trivial local fundamental groups (see [4]).

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.

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.

Is it ethical to prolong life by providing aggressive palliative care to chronically ill patients with a short life expectancy? Here, Andrea Cavicchioli argues that epidemiology studies can help us to resolve this difficult ethical dilemma.

In this paper we study algebraic and geometric properties of closed oriented smooth 4-manifolds M with H2(M;Z)≅0. Moreover, we investigate the problem of embedding M in 5-space or other standard simply-connected 5-manifolds according to Barden's list [Ann. of Math. 82 (1965) 365–385]. These results are related with papers [Invent. Math. 77 (1984) 173–184; Topology 23 (1984) 257–269] of Cochran.

We deal with three combinatorial representations of closed orientable 3-manifolds, i.e., Heegaard diagrams, branched coverings, and crystallizations (a special class of pseudo-graphs endowed with proper edge-colorings). Exploring the connections between those theories, we prove the validity of a conjecture, stated by Dunwoody in [14], concerning the class of closed orientable 3-manifolds represented by symmetric Heegaard diagrams. As a consequence, we classify the topological and geometric structures of many interesting classes of cyclic branched coverings of (hyperbolic) knots encoded by cyclic presentations of groups. In all cases, we show that the polynomial associated with the cyclic presentation coincides (up to a multiplicative unit) with the Alexander polynomial of the considered knot. Finally, we include a partial output of a computer program which generates symmetric Heegaard diagrams of cyclic branched coverings of 3-bridge knots up to nine crossings.