The classical Godement Theorem for manifolds, characterizing kernel pairs for submersions (cf. e.g. [15], LG IV §5), has been used by Pradines[12] as a crucial property for having a good theory of differentiable groupoids; in fact, he developed an axiomatic theory of categories in which Godement's and some other exactness properties hold, under the name of ‘Godement diptych’.

Throughout this note, A will denote a (commutative, Noetherian) local ring (with identity) having maximal ideal m and dimension d. Let x1, …, xd be a system of parameters (s.o.p.) for A. A (not necessarily finitely generated) A-module M is said to be a big Cohen–Macaulay A-module with respect to x1, …, xd, if x1, …, xd is an M-sequence. In the last ten or fifteen years there has been substantial interest in such modules, initially stemming from M. Hochster's discoveries that, if A contains a field as a subring, and x1, …,xd is any s.o.p. for A, then there exists a big Cohen-Macaulay A-module with respect to x1, …,xd, and that the existence of such modules has important consequences for the local homological conjectures in commutative algebra: see [6].

Although this paper is concerned with answering a question in general topology about sequential convergence, the techniques used may be of greater interest to those interested in the structure of filters on ω. A point in a topological space is called an α1-point if whenever is a countable family of sequences converging to it, there is a sequence B, also converging to it, such that A\B is finite for each A ∈ . A space is α1 if each point of the space is an α1-point. Nyikos has shown that the existence of countable Fréchet α1-spaces which are not first countable follows from the assumption and has asked if it is consistent that no such space exists. It is not difficult to see that if there is an α1-space which is not first countable the there is also one with only one non-isolated point. Since, in answering Nyikos' question, we are only concerned with countable spaces, it follows that we are really dealing with certain types of filters on ω. The precise translation of the topological question to a filter theoretic one is contained in §2. The reader who is not interested in the details may wish to read only §2, the first half of §3 and §5.

Dense immersions occur frequently in Lie group theory. Suppose that exp: g → G denotes the exponential function of a Lie group and a is a Lie subalgebra of g. Then there is a unique Lie group ALie with exponential function exp:a → ALie and an immersion f:ALie→G whose induced morphism L(j) on the Lie algebra level is the inclusion a → g and which has as image an analytic subgroup A of G. The group Ā is a connected Lie group in which A is normal and dense and the corestriction

is a dense immersion. Unless A is closed, in which case f' is an isomorphism of Lie groups, dim a = dim ALie is strictly smaller than dim h = dim H.

It is shown that under certain general Tauberian conditions the asymptotic relationship

between two Dirichlet series implies the asymptotic relationship

In a recent paper [6] the authors introduced the concept of a completely 0-simple semigroup of quotients. This definition has since been extended to the class of all semigroups giving a definition of semigroup of quotients which may be regarded as an analogue of the classical ring of quotients. When Q is a sernigroup of quotients of a semigroup S, we also say that S is an order in Q.

Let Σg be a closed orientable surface of genus g, which will be assumed to be greater than one throughout this paper. In our previous paper [11], we have associated to any oriented ∑g-bundle π: E → X with a cross-section s: X → E a flat T2g-bundle π: J → X and a fibre-preserving embedding j: E → J such that the restriction of j to any fibre Ep = π−1(p)(p ∈ X) is equivalent to the Jacobi mapping of Ep with respect to some conformal structure on it and relative to the base-point s(p) ∈ Ep. There is a canonical oriented S1-bundle over J and the main result of [11] is the identification of the Euler class of the pullback of this S1-bundle by the map j as an element of H2(E, ). In this paper we deal with the case of Σg-bundles without cross-sections. First of all we associate a flat T2g-bundle π′: J′ → X to any oriented Σg-bundle π: E → X and construct a fibre-preserving embedding j′: E → J′ such that the restriction of j′ to any fibre Ep is equivalent to some translation of the Jacobi mapping of it. Although our original motivation for the present work came from a different source, this should be considered as a topological version of Earle's embedding theorem [5] which states that any holomorphic family of compact Riemann surfaces over a complex manifold can be embedded in a certain associated family of the corresponding Jacobian varieties in an essentially unique way.

We give a combinatorial characterization of the known examples of intervals of ℤ having the Littlewood-Paley (LP) property. This leaves an interesting open problem. We also note that certain previously known necessary conditions are equivalent, and give two examples of intervals which are not LP but whose endpoints form thin sets (e.g. Sidon or Λ(p) for all p < ∞).

A band is a semigroup in which every element is idempotent. In this note we give an explicit description of the Jacobson radical of the semigroup ring of a band over a ring with unity. It is shown, further, that this radical is nil if and only if the Jacobson radical of the coefficient ring is nil. For the particular case of a normal band (see below for the definition) the Jacobson radical of the band ring is nilpotent if and only if the Jacobson radical of the coefficient ring is nilpotent; but this result does not extend to arbitrary bands.

The polynomial invariants of classical links that relate to the Jones polynomial have various inter-connections. Here a new one is given: the square of the Jones polynomial is obtained from the F-polynomial by means of a certain substitution of variables. This explains and corrects some similarities between known evaluations of these polynomials. The proof uses the ‘Dubrovnik’ version of the F-polynomial; this is first proved to be equivalent to the F-polynomial. Finally a correlation between the F and P polynomials is included as a provocative curiosity.

In [1], J. Bruna studied the Beurling classes EM(I) of infinitely differentiable functions f defined on an interval I such that for every positive ε, there exists a constant Cε > 0 with the property that

where M = {Mn} is a given sequence of positive numbers. With the hypothesis that the class EM(ℝ) is differentiable, he proved that it is inverse-closed (in the sense that if fεEM(ℝ) and if f is bounded away from zero on ℝ, then its inverse f-1 lies in EM(ℝ)) if and only if the associated sequence A = {An = (Mn/n!)1/n} is almost increasing (i.e. Am ≤ KAn for all m ≤ n, where K > 0 is a constant independent of m and n).

The aim of this paper is to discuss categories arising when we extend the metric structure to pointless spaces (frames or locales). The classical distance function is here being replaced by the notion of diameter satisfying certain natural properties (monotonicity, subadditivity for intersecting elements, etc.).

The 2-jet of a Σ3 map-germ f:(3, 0) → (3, 0) determines a net of quadratic maps from 3 to 3; for nets of general type this jet is sufficient for equivalence. The classification of such nets involves a single parameter c. It is shown in [7], also in [3], that the versai unfolding of f is topologically trivial over the parameter space. However, there are 4 connected components of this space of nets. The main object of this paper is to show that the corresponding unfolded maps are of different topological types.

Let S(ℝn) be the space of Schwartz class functions and S'(ℝn) be the dual space of S(ℝn). Given a k-dimensional manifold M in ℝn with area (in case k = 1, M is a curve with length), then for 1 ≤ p ≤ ∞ we say that M has the p-approximate property of for each TεS'(ℝn) with supp T⊆M, and εLp(ℝn), we can find a sequence of measures {Tj,j = 1,2,…} on M, absolutely continuous with respect to the area measure on M, such that ∥j-∥p → 0 as j → ∞.

Several important classes of Banach spaces are characterized by means of convergence properties of sequences. For example, if X is a Banach space, then X belongs to the class Nl1 of spaces without copies of l1, the class R of reflexive spaces or the class F of finite-dimensional spaces if and only if each bounded sequence has respectively a weakly Cauchy (w-Cauchy), weakly convergent (w-convergent) or convergent subsequence. Similarly X is in the class WSC of weakly sequentially complete spaces, or the class SCH of spaces with the Schur property if and only if each w-Cauchy sequence is w-convergent, or convergent, respectively; note that X ∈ SCH if and only if each w-convergent sequence of X is convergent (see [12], p. 47).

A tangle is a pair of strings (t0, t1) properly embedded in a 3-ball. Tangles have been used in several approaches to the classification of knots (see [1, 4, 15]). In these investigations, one keeps track of the endpoints of the arcs, so that the sum of two tangles along their boundaries is well defined. In particular, the sum of a given tangle with a trivial tangle, and any invariants of the resulting link, are invariants of the tangle under the restricted relation of isotopy keeping the endpoints fixed.

We are concerned in this paper with polynomials in n variables xi over the field 2 of two elements. The Steenrod squares Sqk are linear operators which act on the variables by the rules

and on the product uv of homogeneous polynomials u and v by the Cartan formula

1. In a recent paper [6], the spectrum of the Cesàro operator C on c0 (the space of null sequences of complex numbers with the sup norm) was obtained by finding the eigenvalues of the adjoint operator on and showing that the operator (C–λI)−1 lies in B(c0) for all λ outside the closure of this set of eigenvalues. In this paper we apply a similar method to find the spectrum of the two-dimensional Cesàro operator on a space of double sequences c0(c0) (defined in §2). We shall introduce a simplification to the proof in [6] by observing that (C – λI)−1, when it exists, is a Hausdorff summability method (see page 288 of [11] for the single variable case on the space of convergent sequences c), and the crux of our proof is to show that the moment constant associated with the method (C – λI)−1 is regular for the space c0(c0) and the set of λ under consideration. It turns out that c0(c0) ≅ c0c0 (see page 237 of [7]) and that the two-dimensional Cesàro operator on c0(c0) is the tensor product C ⊗ C of the Cesàro operator C on c0. Thus our result gives a direct proof that the spectrum σ(C ⊗ C) equals σ(C)σ(C), which is a special case of the result of Schechter in [8].

Let X denote a Banach space over the complex field ℂ and let B(X) be the Banach algebra of all bounded linear operators on X. If T ε B(X), we write n(T) = dim ker (T) and d(T) = codim T(X). Suppose that Y is a subspace invariant under T; then TY will denote the restriction of T to Y and Y the operator on X/Y defined by

Y: x/Y →(Tx)/Y

R. M. W. Wood has proved a conjecture we made about dimensions of A generators of H*(RP∞ × … × RP∞). In this note we give some of the consequences of this conjecture.