The compact–open topology for function spaces is usually attributed to R. H. Fox in 1945 [16]; and indeed, there is no earlier publication to attribute it to. But it is clear from Fox's paper that the idea of the compact–open topology, and its notable success in locally compact spaces, were already familiar. The topology of course goes back to Riemann; and to generalize to locally compact spaces needs only a definition or two. The actual contributions of Fox were (1) to formulate the partial result, and the problem of extending it, clearly and categorically; (2) to show that in separable metric spaces there is no extension beyond locally compact spaces; (3) to anticipate, partially and somewhat awkwardly, the idea of changing the category so as to save the functorial equation. (Scholarly reservations: Fox attributes the question to Hurewicz, and doubtless Hurewicz had a share in (1). As for (2), when Fox's paper was published R. Arens was completing a dissertation which gave a more general result [1] – though worse formulated.)

In the paper ‘Bordism and Cobordism’, which appeared in vol. 57 (1961) of this journal [5], Michael Atiyah introduced and began the study of bordism and cobordism theory. The present article will trace developments in this area since this beginning.

Sturm's theorem was announced in 1829 and various proofs were given soon after (see [2], p. 363 for details). It is given in various textbooks on algebra, e.g. in [1], pp. 448–453, where its application to finding the number and location of the real zeros of a polynomial is explained. It has been applied, in conjunction with Givens's method, to finding the eigenvalues of a matrix (see, e.g. [3], p. 494), and has been used recently in a neat computational form by Takeuchi [4] for investigating algebraic fields.

Let λ(n) denote the Liouville function, i.e. λ(n) = 1 if n has an even number of prime factors, and λ(n) = − 1 otherwise. It is natural to expect that the sequence λ(n) (n ≥ 1) behaves like a random sequence of ± signs. In particular, it seems highly plausible that for any choice of εi = ± 1 (i = 0,…, k) we have

Let K be an algebraic number field and f(X) ∈ K[X]. The Diophantine problem of describing the solutions to equations of the form

has attracted considerable interest over the past 60 years. Siegel [12], [13] was the first to show that under suitable non-degeneracy conditions, the equation (+) has only finitely many integral solutions in K. LeVeque[7] proved the following, more explicit, result. Let

where a ∈ K* and αl,…,αk are distinct and algebraic over K. Then (+) has only finitely many integral solutions unless (nl,…,nk) is a permutation of one of the n-tuples

In this paper we have two objectives. The first is to investigate the restriction of a metaplectic cover to an arbitrary torus in GL2. This will be explained at greater length below, and the main results are Theorems 1 and 2. The second is an application of the same ideas to introduce the arithmetic function P, which has already appeared in a special case in [9], and to prove the fundamental property given by Theorem 3. These theorems will be proved in §§ 2 and 3. In §§ 4 and 5 we remark on the appearence of the function P in the formula of Loxton and Matthews [5], [6] for the biquadratic Gauss sum and discuss the structure of this formula.

A family of sets is said to possess property B if there is a set T such that A ∪ T ╪ ø and yet A ⊈ T for all sets A in . These two requirements may be restated in terms of the characteristic functions of the sets in as follows:

Throughout, p will be a fixed prime, and will denote the class of all locally finite p-groups. For a fixed Abelian p-group A, we let

where ζ(P) denotes the centre of P. Notice that A is not a class in the usual group-theoretic sense, since it is not closed under isomorphisms.

The graphs we consider here are either simple graphs, that is they have no loops or multiple edges, or are multigraphs, that is they may have more than one edge joining a pair of vertices, but again have no loops. In particular we shall consider a special kind of multigraph, called a star-multigraph: this is a multigraph which contains a vertex v*, called the star-centre, which is incident with each non-simple edge. An edge-colouring of a multigraph G is a map ø: E(G)→, where is a set of colours and E(G) is the set of edges of G, such that no two edges receiving the same colour have a vertex in common. The chromatic index, or edge-chromatic numberχ′(G) of G is the least value of || for which an edge-colouring of G exists. Generalizing a well-known theorem of Vizing [14], we showed in [6] that, for a star-multigraph G,

where Δ(G) denotes the maximum degree (that is, the maximum number of edges incident with a vertex) of G. Star-multigraphs for which χ′(G) = Δ(G) are said to be Class 1, and otherwise they are Class 2.

This paper treats two related sets of problems in the theory of random graphs. In Sections 2 and 3 we study random spanning subtrees of a complete graph (or, equivalently, random labelled trees). It is shown that the number of common edges of two such random trees asymptotically has a Poisson distribution with expectation 2. Similar results are obtained for the number of edges in the intersection or union of more than two random trees.

In 1979 at the Siegen Topology conference Peter Eccles in his lecture asked and in most cases answered the following

Question. For which values of n does an immersion of a closed n-dimensional manifold into Rn+1 exist with a single (n + 1)-tuple point?

The answer (see [3–5, 8]) implies the following:

Proposition (Eccles). No immersion of an even dimensional orientable manifold Mn into Rn+1 has a single (n + l)-tuple point.

In this paper we prove a theorem which gives general conditions under which the free abelian topological group F(Y) on a space Y can be embedded in the free abeian topological group F(X) on a space X.

In ([2], p. 310) the problem of relating the Arf invariant of a link to other link invariants, and finding practical ways of computing it, was raised. Murasugi gave a partial solution to this in [9]: if L has two components, then Arf(L), when defined, is given by

where l1, l2 are the components of L, and ΔL(t1, t2) denotes the Alexander polynomial of L.

There exists for each oriented link L in the 3-sphere a unique 2-variable Laurent polynomial defined uniquely by the following:

(i) PU = 1 for the unknot U

(ii) ,

where L+, L− and L0 are identical outside a ball and inside are as shown in Figure 1. See [1] and [2].

We recall a theorem of Hurwitz [5]: let f(θ) be real-valued and periodic, f(θ) ≢ 0, and

where an = bn = 0, 0 ≤ n < N. Then f(θ) has at least 2N sign changes in each period. A similar result with a trigonometric polynomial on the right-hand side of (1) had been obtained earlier by Sturm [7].

In this paper we present two versions in several variables of the following result:

Theorem 1([2, 3]). Let f be a function in the disc algebra (more generally in H1). Then for every point z0 in the open unit disc, there is an interval I on the unit circle T such that f(z0) = 1/|I| ∫Ifdσ, where 0 < |I| ≤ 2π denotes the length of I and σ the Lebesgue measure on T.