Reshetikhin and Turaev [10] gave a method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra (e.g. a quantum group Uq(sl2)) using finite-dimensional representations of it. In this paper we give another independent method to construct a topological invariant of compact oriented 3-manifolds from a ribbon Hopf algebra via universal invariants of framed links without using representations of the algebra. For Uq(sl2) these two methods give different invariants of 3-manifolds.

A formal series Σxn in a Banach space X is said to be weakly unconditionally converging, or alternatively weakly unconditionally Cauchy (wuc) if Σ|x*(xn)| < ∞ for every continuous linear functional x* ∈ X*. A subset K of X* is called a V-subset of X* if

for each wuc series Σxn in X. Further, the Banach space X is said to have property (V) if the V-subsets of X* coincide with the relatively weakly compact subsets of X*. In a fundamental paper in 1962, Pelczynski [10] showed that the Banach space X has property (V) if and only if every unconditionally converging operator with domain X is weakly compact. In this same paper, Pelczynski also showed that all C(Ω) spaces have property (V), and asked if the abstract continuous function space C(Ω, X) has property (F) whenever X has property (F).

One of the outstanding problems in four-dimensional topology is to find the minimal genus of an oriented smoothly embedded surface representing a given homology class in a smooth four-manifold. For an arbitrary homology class in an arbitrary smooth manifold not even a conjectural lower bound is known. However, for the classes represented by smooth algebraic curves in (simply connected) algebraic surfaces, it is possible that the genus of the algebraic curve, given by the adjunction formula

is the minimal genus. This is usually called the (generalized) Thom conjecture. It is mentioned in Kirby's problem list [11] as Problem 4·36.

We consider links in an orientable 3-manifold M which have an alternating diagram on a closed orientable surface F of positive genus. We see that if the diagram is ‘complex’ enough and if F gives a Heegaard splitting of M, then such a link L is prime and M—L does not contain an essential torus.

In [2] it was conjectured that the coloured Jones function of a framed knot K, or equivalently the Jones polynomials of all parallels of K, is sufficient to determine the Alexander polynomial of K. An explicit formula was proposed in terms of the power series expansion , where JK, k(h) is the SU(2)q quantum invariant of K when coloured by the irreducible module of dimension k, and q = eh is the quantum group parameter.

In this paper I show that the explicit formula does give the Alexander polynomial when K is any torus knot.

If f: X × Y → Z is a map, then the classical Hopf construction associates to f a map hf: X * Y → ΣZ, where X * Y is the join of X and Y and ΣZ the suspension of Z. Since X * Y has the homotopy type of Σ(X Λ Y), the suspension of the smash product of X and Y, the homotopy class of hf can be regarded as an element Hf ↦ [Σ(X Λ Y), ΣZ]. Now elements of [Σ(X Λ Y), ] are in one to one correspondence with homotopy classes in the group [σ(X Λ Y), ΣZ] which are trivial on the suspension of the wedge Σ(X ≷ Y).

If A is a complex Banach algebra the socle, denoted by Soc A, is by definition the sum of all minimal left (resp. right) ideals of A. Equivalently the socle is the sum of all left ideals (resp. right ideals) of the form Ap (resp. pA) where p is a minimal idempotent, that is p2 = p and pAp = ℂp. If A is finite-dimensional then A coincides with its socle. If A = B(X), the algebra of bounded operators on a Banach space X, the socle of A consists of finite-rank operators. For more details about the socle see [1], pp. 78–87 and [3], pp. 110–113.

This paper may be regarded as a sequel to our earlier paper [19], where we give an elementary and self-contained proof of a very general form of the Hopf theorem on order-preserving linear operators in partially ordered vector spaces (reproduced here as Theorem 1·1).

Versions of this theorem and related ideas have been used by various authors to study both linear and nonlinear integral equations (Thompson [41], Bushell [9, 11], Potter [38, 39], Eveson [16, 17], Bushell and Okrasiriski [12, 13]); the convergence properties of nonlinear maps (Nussbaum [32, 33]); so-called DAD theorems (Borwein, Lewis and Nussbaum [8]) and in the proof of weak ergodic theorems (Fujimoto and Krause [20], Nussbaum [34]).

Let G be a finite group of orientation preserving isometrics of a closed orientable hyperbolic 2-manifold Fg of genus g > 1 (or equivalently, a finite group of conformal automorphisms of a closed Riemann surface). We say that the G-action on Fgbounds a hyperbolic 3-manifold M if M is a compact orientable hyperbolic 3-manifold with totally geodesic boundary Fg (as the only boundary component) such that the G-action on Fg extends to a G-action on M by isometrics. Symmetrically we will also say that the 3-manifold M bounds the given G-action. We are especially interested in Hurwitz actions, i.e. finite group actions on surfaces of maximal possible order 84(g — 1); the corresponding finite groups are called Hurwitz groups. First examples of bounding and non-bounding Hurwitz actions were given in [16].

Equivariant shape theory is an improvement of equivariant homotopy theory which could be regarded as an equivariant version of Borsuk's shape theory. The main result in this paper is a description of the equivariant shape category ShG whose objects are equivariant spaces or G-spaces, i.e. topological spaces endowed with an action of a given topological group G, and whose morphisms are equivariant homotopy or G-homotopy classes of families of multi-valued functions which we call equivariant multi-nets or G-multi-nets. Previously equivariant shape theories have been described only under the assumptions that the group G is either finite or compact. We also study classes of G-spaces on which equivariant shape and equivariant homotopy coincide, look for conditions under which a G-map f: X → Y is an equivariant shape equivalence, and give some characterizations of G-spaces with trivial equivariant shape.

Let D be the open unit disc in the complex plane C and let dA be the normalized area measure on D. The Bergman space is the space of analytic functions f in D such that

Let S(Rn) be the space of Schwartz class functions. The dual space of S′(Rn), S(Rn), is called the temperate distributions. In this article, we call them distributions. For 1 ≤ p ≤ ∞, let FLp(Rn) = {f:∈ Lp(Rn)}, then we know that FLp(Rn) ⊂ S′(Rn), for 1 ≤ p ≤ ∞. Let U be open and bounded in Rn−1 and let M = {(x, ψ(x));x ∈ U} be a smooth hypersurface of Rn with non-zero Gaussian curvature. It is easy to see that any bounded measure σ on Rn−1 supported in U yields a distribution T in Rn, supported in M, given by the formula

In the study of polynomials acting on Banach spaces, the weak topology is not such a good tool as in the case of linear operators, due to the bad behaviour of the polynomials with respect to the weak convergence. For example,

is a continuous polynomial taking a weakly null sequence into a sequence having no weakly Cauchy subsequences. In this paper we show that the situation is not so bad for unconditional series. Recall that is a weakly unconditionally Cauchy series (in short a w.u.C. series) in a Banach space E if for every f ε E* we have that and is an unconditionally converging series (in short an u.c. series) if every subseries is norm convergent.

Let U be an invariant component of the Fatou set of an entire transcendental function f such that the iterates of f tend to ∞ in U. Let P(f) be the closure of the set of the forward orbits of all critical and asymptotic values of f. We show that there exists a sequence pn∈P(f) such that dist(pn, U) = o(|pn|), where dist(·, ·) denotes Euclidean distance. On the other hand, we give an example where dist (P(f), U) > 0. In this example, U is bounded by a Jordan curve.

Let S be a normal projective algebraic surface over C with at worst quotient singularities. S is a quasi-log del Pezzo surface if the anti-canonical divisor — Ks is nef (= numerically effective) and big, i.e. — Ks. C ≥ 0 for all curves C on S and (−Ks)2 > 0. Further, if — Ks is ample we say S is a log del Pezzo surface.

Let ρ,ρ0,ρ1 be positive, measurable functions on ℝN. For 1 ≤ t < ∞, consider the weighted Lebesgue and Sobolev spaces

Let B(r) denote the open ball of radius r centred at the origin 0 of the Euclidean space ℝN, where N ≥ 2. It is well known that if h is harmonic in B(1), then there exist homogeneous harmonic polynomials Hj of degree j in ℝN such that converges absolutely and locally uniformly to h in B(1) (see, e.g. Brelot[1], Appendice). Further, this series is unique and each Hj is the sum of all the monomial terms of degree j in the multiple Taylor series of h centred at 0. We call the polynomial expansion of h.

For ξ ε (0, ½) let Eξ be the perfect symmetric set of constant ratio ξ and set

It was shown by the first author that if T is a contraction on the Hilbert space H with spectrum contained in Eξ, and if log ∥T−n∥ = O(nα) as n → ∞ for some α < b(ξ), then T is unitary. In the other direction, we show here that there exists a (non-unitary) contraction T on H such that SpT = Eξ, log ∥T-n∥ = O(nb(ξ)) as n → ∞, and lim supn → ∞ ∥T−n∥ = ∞.

Let ψ be a distribution function on [−1,1] from the Szegö-class, which contains in particular all Jacobi weights, and let (pn) be the monic polynomials orthogonal with respect to dψ. Let m(n)∈ℕ, n∈ℕ, be non-decreasing with limn → ∞ (n − m(n)) = ∞, l(n)∈ℕ with 0 ≤ l(n) ≤ m(n), and μj, n ∈ℝ for j = 0, …, m(n), n∈ℕ. It is shown that for each sufficiently large n, has n−l(n) simple zeros in (−1, 1)and l(n) zeros in ℂ\[−1,1] if for n ≥ n0, has m(n) − l(n) zeros in the disc |z| ≤ r < 1, l(n) zeros outside of the disc |z| ≥ R > 1 and where q > 2 max {r, 1/R}. If m(n) is constant for n ≥ n0 then the statement holds even for such polynomials (pn) orthogonal with respect to a distribution dψ satisfying the weak assumption ψ′ > 0 a.e. on [−1, 1]. For linear combinations of polynomials orthogonal on the unit circle corresponding results are derived.

Consider a knot K in a closed, oriented 3-manifold M. A genus g Heegaard decomposition of M is said to be a genus g bridge decomposition of K if it also decomposes K into trivial arcs in each handlebody. The genus g bridge number of K, denoted by bg(K), is defined as the minimal number of trivial arcs in a handlebody among all the genus g bridge decompositions of K [2, 13].