Let X be a CW-complex with finite skeletons and with H*(X, Z) free of p-torsion, where p is some odd prime. We shall prove two theorems on the action of the Steenrod powers

1. Let f(t) be a perodic function with period 2π and integrable in the sense of Lebesgue over (–π, π), then its Fourier series is

Let us denote by k*( ) the homology theory determined by the connective BU spectrum, bu, that is, in the notations of (1) and (9), bu2n = BU(2n,…,∞), bu2n+1 = U(2n + 1,…, ∞) with the spectral maps induced via Bott periodicity. The resulting spectrum, bu, is a ring spectrum. Recall that k*(point) ≅ Z[t], degree t = 2. There is a natural transformation of ring spectra

inducing a morphism

of homology functors. It is the objective of this note to establish: Theorem. Let X be a finite complex. Then there is a natural exact sequence

where Z is viewed as a Z[t] module via the augmentation

and, is induced by η*in the natural way.

In (3), the author defined the notion of a G-space. A G-space is a weaker notion than that of an H-space. The main purpose of this paper is to present various means of constructing G-spaces. As an application of some of the techniques of (3) and of this paper (though not an application of the concept of G-space) we shall prove the following theorem:

Theorem. Let G be a connected compact Lie group and let H be a connected subgroup of maximal rank. Then H3(G/H; Z) = 0. In fact, the Hurewicz homomorphism is trivial for odd dimensions.

Let M and Q be closed manifolds, and write ℝ+ = [0, ∞). Our main result is that, in both the PL and topological categories, M × ℝ+ unknots in Q × ℝ+ in codimension ≥ 3 (see Theorem 1 for a precise statement). The proof is essentially a generalization of Stallings's topological unknotting of spheres(13), the main tool being engulfing.

Introduction. We will study here for normal T1 spaces the space of cuts—defined and analysed by Hurewicz (3) for separable metrizable spaces.

The conformal tensors of a Riemannian space Vn are classified and analysed, mainly in terms of the ‘conformal number’ χ which is defined as a certain linear combination of the numbers that characterize the transformation modes of a conformal tensor. In the case of zero rest mass fields (5), χ – 1 is just the spin. Conformal concomitants of the metric tensor and its derivatives up to a finite order m are considered and, in particular, those tensors which are polynomials in the 2nd and higher derivatives of the metric tensor (called conformal P-tensors of order m). The algebraic structure of such a tensor is described by a set of m − 1 integers (S2, …, Sm), referred to as the structure of F and defined in an invariant way in terms of the various polynomial degrees of F and its partial derivatives. The structure of F is related to the m transformation modes of F by the formula: . Thus for P-tensors, χ = 0, 2, 3, … and m ≤ χ. The set of structures (S2, …, Sm) of all conformal P-tensors of order less than or equal to m, forms an Abelian monoid m, with addition defined componentwise. In a general V4, it is found that 4 consists of all triplets (S2, S3, S4) ∈ 3 for which S3 ≤ 4S2. Finally, new conformal P-tensors are constructed so that examples can be given corresponding to every structure in 4. One also obtains, for each value of χ, a general formula which expresses any conformal P-tensor with conformal number χ in terms of a standard sequence of conformal P-tensors.

In the general theory of relativity a number of apparently unrelated identities peculiar to a 4-dimensional space are frequently used. However, the proofs usually presented appear to have no common ideas and, furthermore, it is not clear at what stage the dimensionality restriction plays a significant role. In this note a technique is presented which explicitly exhibits the dimensionality dependence and thereby enables the above identities to be generalized for n-dimensional spaces. It is also shown that the above identities are all special cases of a more general identity valid for n = 4.

1. Introduction. This note is a sequel to (1), the notation of which will be used without further explanation. In (2), certain deductions from the theorems proved in (1) were made in the case where K(x, t) is derived from a function K(x), a.c. in [0, 1], by the equation

All spaces considered in this paper are Hausdorff. θX is a compactification of a completely regular space X means that X is identified with a dense subspace of θX. We shall thus always regard X to be contained in θX. A compactification θX of X is zero-dimension (countable) if its ‘outgrowth” θX ~ X is zero-dimensional (countable).

It is the purpose of this paper to prove a general compactness theorem and deduce from it, as corollaries, the Tychonoff theorem on the product of compact spaces and Ascoli-type theorems giving sufficient conditions for function spaces with the topology of uniform convergence on compact sets to be compact.

A function f( , ) of two variables is said to be separately continuous if for each fixed y and each fixed x, the functions f(, y) and f(x,) are continuous functions of one variable. Separately continuous functions seem to be playing a larger rôle in analysis than formerly. In (1) there is an account of the theory of semigroups in which the multiplication is separately continuous, and in (9) various analytic theorems are obtained by the study of separately continuous functions f(s, t), where t ranges over a set T, and s ranges over a set of functions on T.

In (2) the author determined a formula for computing, by means of the Eilenberg-Moore spectral sequence, the action of the group in a principal G-bundle, under the hypothesis that H*(G) be universally transgressively generated. This rather severe restriction was due to the method of proof, which used the theory of Guy Hirsch. Since Hirsch's work can be derived as a special case of Brown's twisted tensor products (3), one might hope to obtain the formula for the group action in the general case from the latter theory, and this we shall do (Theorem 1·2 below).

It has been pointed out to me by Professor A. M. Macbeath that the proof of Theorem 2 which is given in (2) is incorrect, in that it relies on Proposition 3·1 of (2), which is false, the mistake in its proof being the assumption that in an arbitrary compact topological space, every sequence has a convergent subsequence. However, we can still prove Theorem 2 of (2), simply by replacing section 3 of (2) by the corrected section 3 below, and replacing the proof of Proposition 4·2 of (2) by the corrected proof of Proposition 4·2 given below.

Sawon and Warzecha established in (2) the general form of a subalgebra of codimension 1 in a commutative Banach algebra with identity. A short and elegant proof of their result was later given by Detraz in(1). This note is intended to show that an even more elementary proof is possible and which has the advantage that no use is made of the existence of maximal ideals.

1. Introduction. In his papers (2), (3) and (4), and in his book (5), Husain studied the locally convex, Hausdorff, linear topological spaces (hereafter abbreviated to l.c. spaces) which he called B() spaces and Br() spaces and which satisfy weakened forms of the B completeness and Br completeness conditions of Ptak (8). L.c. spaces satisfying another weakening of the Br completeness condition were studied by Persson in (7) and were called by him, weakly t-polar. We consider some examples in connexion with these spaces.

Let A(Z) be the Banach algebra of Fourier-Lebesgue coefficients: a function f is an element of A(Z) if it is the Fourier transform

of some function fεL1(T), where T, as usual, denotes the circle group. We write

Let G be a locally compact, unimodular, topological group with μ Haar measure on G, and μ* the corresponding inner measure. If (G) denotes the Borel subsets of G of finite measure, and V(G) = {μ(E):E∈(G)}, then the Product Set function, or P.S.-function, of the group G, written ΦG, is defined by

Let A = (ank) be an infinite matrix of complex numbers ank (n, k = 1, 2,…) and X, Y two subsets of the space s of complex sequences. We say that the matrix A defines a (matrix) transformation from X into Y, and we denote it by writing A: X → Y, if for every sequence x = (xk)∈X the sequence Ax = (An(x)) is in Y, where An(x) = Σankxk and the sum without limits is always taken from k = 1 to k = ∞. The sequence Ax is called the transformation of x by the matrix A. By (X, Y) we denote the class of matrices A such that A: X → Y.

We show that under quite general circumstances a Banach algebra R need not be closed in its tilda algebra . In answer to a query of Katznelson we show that there is a very strongly homogeneous algebra B ≠ C0(R) on R with R as carrier space such that non-analytic functions operate on B.