Let be a closed 3-braid in S3 with axis Lγ. The exterior of γ U Lγ in S3 is a fibre bundle over S1 with fibre a disc with 3 holes. The monodromy is given by a matrix in PSL(2,) and the braid γ is called hyperbolic if its matrix is hyperbolic.

The cohomology of finitely generated metabelian groups has been studied in a series of papers by Bieri, Groves, and Strebel [2, 3, 4]. In particular, Bieri and Groves [2] have shown that every metabelian group of type (FP)∞ is of finite rank, and so is virtually of type (FP). The purpose of the present paper is to provide a generalization of this result and to use our methods to prove the existence of a pathological class of finitely generated soluble groups.

An almost tangent structure on a manifold N is a type (1,1) tensor field S onN with the property that at each point y ∊N the kernel of Sy (regarded as a linear endo-morphism of TyN) coincides with its image. An almost tangent structure is said to be integrable if its Nijenhuis tensor vanishes.

In [5] we proved two results: theorem 1, which said that if k was a simple (2q – 1)-knot, q 1, then it was equivalent to the m-fold branched cyclic cover of another knot if and only if there existed an isometry u of its Blanchfield pairing 〈,〉, whose mth power was the map induced by a generator t of the group of covering translations associated with the infinite cyclic cover of k; and theorem 2, which showed that if k were the m-fold b.c.c. of two such knots, then these would be equivalent if and only if the corresponding isometries were conjugate by an isometry of 〈,〉. Using this second result, we present two cases where k may only be the m-fold b.c.c. of finitely many knots.

In the early sixties Effros[9] and Prosser[14] studied, in independent work, the duality of the faces of the positive cones in a von Neumann algebra and its predual space. In an implicit way, this work was generalized to certain ordered Banach spaces in papers of Alfsen and Shultz [3] in the seventies, the duality being given in terms of faces of the base of the cone in a base norm space and the faces of the positive cone of the dual space. The present paper is concerned with the facial structure of the unit balls in an ordered Banach space and its dual as well as the duality that reigns between these structures. Specifically, the main results concern the sets of norm-exposed and norm-semi-exposed faces of the unit ball V1 in a GL-space or complete base norm space V and the sets of weak*-exposed and weak*-semi-exposed faces of the unit ball in its dual space V* which forms a unital GM-space or a complete order unit space.

Let G be a group such that

(1) G is finitely presented with deficiency one,

(2) G/G' is infinite cyclic, with a distinguished generator t.

The axiom of uniform visibility on simply connected manifolds was introduced in [8] by Eberlein and O'Neill. Using the axiom, they were able to extend to a class of simply connected manifolds of non-positive sectional curvature many results which were known to be true for the case of strictly negative curvature. The axiom proved very useful in obtaining results about the geodesic flow on manifolds of non-positive sectional curvature [3], [4].

In this paper we present a proof of G. W. Whitehead's conjecture about symmetric products of the sphere spectrum S. Our methods are based on the transfer, the Steinberg module, and the structure of the Hecke algebra. Our results are valid for all primes and extend those of the first author for p = 2 [7]. As originally stated, the conjecture asserts that

is zero on p-primary components in positive degrees [11]. By considering the quotients L(k) = Σ-kSPpkS/SPk-1-S, this is seen to be equivalent to the exactness of

on homotopy groups, where ∂k is the connecting map and ε is the inclusion of the bottom cell. Here and throughout all spaces and spectra are localized at p.

In the theory of locales (or ‘pointless topology'), several authors have tried to find a suitable form of a T1-separation axiom. Their proposals are unsatisfactory in the sense that they do not coincide for topological spaces with the T1-axiom. Our main result is that sublocales of sober T1-spaces are exactly locales in which primes are dual atoms. Hence, these locales should be called T1, despite the fact that they include any locale without points. Our T1-locales are also closed with respect to products; in fact they form the smallest epireflective subcategory of locales containing all sober T1-spaces. Besides T1-locales we will also consider T2-locales and epireflective sub-categories of locales in general.

Let X be a compact Hausdorff space, let E be a (real or complex) Banach space, and let C(X, E) stand for the Banach space of all continuous E-valued functions defined on X under the supremum norm. If A is an arbitrary linear subspace of C(X, E), then it is shown that each bounded linear functional l on A can be represented by a boundary E*-valued vector measure μ on X that has the same norm as l. This result constitutes an extension to vector-valued functions of the so-called analytic version of Choquet's integral representation theorem.

In this paper we shall consider the class Hm of subharmonic functions u, unbounded in, which have a thick tract where u is bounded. We recall here that non-constant subharmonic functions in space (i.e. m ≥ 3) are not necessarily unbounded as in the two-dimensional case. For basic concepts in this area we refer the reader to [3].

For a principal fibre bundle (p, q, B, G) with structure group G, the group of G-equivariant self equivalences of the total space P is investigated by using bundle map theory. Computations are given for well-known principal fibre bundles.

We work in the DIFF category. A knot K = (Sn+2, Sn) is a ribbon knot if Sn bounds an immersed disc Dn+1 →Sn+2 with no triple points and such that the components of the singular set are n-discs whose boundary (n – l)-spheres either lie on Sn or are disjoint from Sn. Pushing Dn+1 into Dn+3 produces a ribbon disc pair D = (Dn+3, Dn+1), with the ribbon knot (Sn+2, Sn) on its boundary. The double of a ribbon (n+1)-disc pair is an (n + l)-ribbon knot. Every (n+l)-ribbon knot is obtained in this manner.

Let V be a smooth, 1-connected 4-manifold. It has long been known that there may be 2-dimensional homology classes in V which are not represented by smoothly embedded 2-spheres, owing to the failure of the Whitney lemma in this dimension. One of the simplest examples of this failure may be described as follows. Let K be a smooth knot in S3 and r an integer. By V(K, r) we mean the 4-manifold obtained by attaching a 2-handle to B4 along an r-framed tubular neighbourhood of K. Evidently each V(K, r) is homotopy equivalent to S2 and we say that K is an r-shake-slice knot if there is a 2-sphere, smoothly embedded in V(K, r), which represents a generator of the second homology group. For example, if K is a slice knot then it is r-shake-slice for each integer r. On the other hand, it is known that arf (K) = 0 whenever K is r-shake-slice and so in particular a trefoil is never such a knot.

Let B(t) be a two-dimensional Brownian motion. For 0 < α < 2π, set

and, for 0 ≥ β< 2π, let F(α,β) be F(α) rotated through an angle β about the origin.

Let Mω(X) be a weighted hypergroup algebra, and Lω(X) be the Banach algebra of measures μ ε Mω(X) such that the function x ↦ (1/ω(x))δx* |μ| is norm continuous. We characterize compact multipliers on Lω(X). This extends the characterization of compact multipliers on weighted group algebras and some classes of weighted semigroup algebras.

There are various ways of using the affine geometry of a compact convex set K to topologize its extreme boundary ∂eK (see [2, 7, 10] and the references quoted therein). The best-known method is to take as closed sets the traces on ∂eK of the closed split faces of K. This topology is known as the facial topology [1, 2, 4], and is significant because the facially continuous functions correspond to the order-bounded linear operators on A(K) and to the multipliers in A(K) [1, 3, 20]. Typically, split faces are scarce, so that the facial topology is coarse. The Choquet topology [7, 15, 18], whose closed sets are the traces of compact extremal subsets of K, is finer, but is usually less easily recognized. The Bishop–de Leeuw theorem was extended in [5] by showing that a maximal measure on K induces a measure on a σ-algebra on ∂eK containing the Choquet closed sets as well as the traces of the Baire subsets of K. The maximal topology is finer even than the Choquet topology, and the Bishop-de Leeuw theorem was further extended in [18] to a σ-algebra including the maximally closed sets. Although the definition of the maximal topology in [18] made use of integral representation theory, a geometrical description of it will be given in Proposition 2·1 below. The Choquet and maximally continuous functions on ∂eK were characterized in [18] in terms of the geometry and integral representation theory of K, respectively. It will be shown in Theorem 2·4 that the Choquet and maximally continuous functions coincide if is a union of faces of K, even though the topologies may differ.

In § 33 of [2], Professor Williams asks whether there are L1 random variables† X, Y, Z (not identically zero) such that

A 2 component link with Alexander polynomial 1 is TOP concordant to the Hopf link. Our argument is modelled closely on Freedman's analysis of the problem of slicing Alexander polynomial 1 knots, and uses his theory of 4-dimensional surgery over groups with polynomial growth. A similar argument shows that certain F-isotopies may be realized by TOP concordances.

An n-knot will be a smooth, oriented submanifold Kn ⊂ Sn+2 such that Kn is homeomorphic to Sn. Given two knots and , we define their connected sum as in [13], p. 39. The cancellation problem for n-knots is the following.