Every oriented knot or link in S3 can be represented in many ways as the closure of a braid β ∈ Bn, the braid group on n strings, for some n. Braids β ∈ Bn, γ ∈ Bm are called closure-equivalent if and are equivalent as oriented knots. It is a well-known result of Markov, see, for example, (l), that β and γ are closure equivalent if and only if there is a sequence of elementary (Markov) moves in which β ∈ Bn is replaced by

(a) a conjugate in Bn, or , or

(c) β1 ∈ Bn−1, where ,

and the process repeated until γ is reached.

Let L be a finite-dimensional Lie algebra and U(L) its universal envelope. It is known that U(L) is a Noetherian domain (see (5), theorem v. 3·4) and therefore U(L) has a field of fractions. (Throughout the paper we use the term ‘field’ in the sense of skew field.) We prove in this article the following theorem.

In (1) it was shown that PL characteristic numbers determine (unoriented) PL bordism. In §§ 1 and 2 of this paper we use D. Sullivan's obstruction theory (9), (5) to provide some geometric insight into this result, as follows. We take an unoriented PL manifold M and let νM: M → BPL classify its stable normal bundle, and assume that all its PL characteristic numbers vanish, i.e. that (νM)* [M] = 0, where [M] is the fundamental class of M and . To show that M bounds, we first ‘resolve the homology to zero’ given by (νM)* [M] = 0. This is done by defining obstructions in whose vanishing ensures that the homology can be ‘resolved’, i.e. can be replaced by a bordism F: W → BPL between νM and a constant map, g: N → BPL, say. Then it is shown that these obstructions all vanish in this case.

This paper addresses the question of how well the exterior of a knot determines the knot. It is well known that for a knotted sphere pair (Sn+2, Sn), n ≥ 2, a given exterior corresponds to at most two distinct knots (1), (4), (10); examples of distinct knots with the same exterior are given in (2), (6). For a knotted ball pair (Bn+2, Bn), n ≥ 2, the situation may be more complicated. In fact, given an arbitrary positive integer N, Hitt-Sumners (7), (8) constructed N distinct ball pairs with the same exterior, n ≥ 4.

We state and prove a theorem which characterizes piecewise linear homology manifolds of sufficiently large dimension among locally compact finite-dimensional absolute neighbourhood retracts (ANRs). The proof is inspired by Cannon's observation (3) that a piecewise linear homology manifold is a topological manifold away from a locally finite set, and uses Galewski and Stern's work on simplicial triangulations of topological manifolds, the Edwards–Cannon–Quinn characterization of topological manifolds and Siebenmann's work on ends (3, 6, 4, 13, 14, 15, 16). All these tools have suitable relative versions and so the theorem can be extended to the bounded case. However, the most satisfactory extension requires a classification of triangulations of homology manifolds up to concordance. This will be given in a subsequent paper and the bounded case will be postponed to that paper.

In (5), we and Fred Cohen gave some quite general splitting theorems. These described how to decompose the suspension spectra of certain filtered spaces CX as wedges of the suspension spectra of their successive filtration quotients Dq X. The spaces CX were of the form Cr × Xr/(˜) for suitable sequences of spaces {Cr} and {Xr}, and the construction CX was intended to be a reworking in ‘proper generality’ of the constructions introduced in (9).

A complete and satisfying account of the classification of pencils of binary cubics over an algebraically closed field was given by Newstead (2). Extending these results to the real case is not a matter of mere routine since new questions arise, for example the separation of roots of the cubics in a pencil (as well as their reality).

Exact sequences, which were subsequently used for calculating the group ℰ(X) of homotopy classes of self-homotopy equivalences of a space X, were given by Barcus and Barratt (§5 of (1)) in the case where X is obtained from a simply-connected (q + 1 > 1)-dimensional complex by adding one (q + l)-cell (q ≥ 3): these were later extended by Kudo and Tsuchida (theorems 2·2 and 2·8 of (6)) and by the author (theorem 3·1* of (15)), who also obtained a related sequence (theorem 2·3* of (15)). In the case of a two-cell complex, one or more of these sequences has been shown to split by Oka, Sawashita and Sugawara (theorems 3·9, 3·13 and 3·15 of (11)). The sequences have been used to calculate ℰ (X) for a number of complexes having two, three or more cells by various authors, including Oka (8), Oka, Sawashita and Sugawara (11), Rutter (17) and Sawashita (18). However the aforementioned sequences are only applicable to the addition of top-dimensional cells if the complex has no cells in its penultimate dimension. In this article I obtain sequences which are applicable without this latter restriction, show that one of them is generally split, and in special cases where there is only one top-dimensional cell obtain a further splitting: sequences are given which are valid without the assumption that A is simply connected. Also I give a new formula for calculating ℰ(X)in the case where X is not 2-connected.

Let K be a cohomology theory. Axioms are given which uniquely characterize the transfer in K-cohomology for bundles with structure group a compact Lie group and fibre a finite C.W. complex. Also, a family of generalized transfers is defined which includes both the standard transfer and Atiyah's holomorphic transfer.

In this note we introduce a general class of finite ramified coverings π X˜ ↓ X. Examples of ramified covers in our sense include: finite covering spaces, branched covering spaces and the orbit map Y ↓ Y/G where G is a finite group and Y an arbitrary G-space. For any d-fold ramified covering π: X˜ ↓ X we construct a transfer homomorphism

with the expected property that

is multiplication by d. As a consequence we obtain a simple proof of the Conner conjecture; viz. the orbit space of an arbitrary finite group action on a ℚ-acyclic space is again ℚ acyclic.

R. C. Kirby and W. B. R. Lickorish have proved (cf. (4)) that any classical knot is concordant to an indecomposable knot. In the present note we show that this statement is also true for higher dimensional knots: more precisely, for any higher-dimensional knot K there exist infinitely many non-isotopic indecomposable simple knots which are concordant to K. This, together with the result of Kirby and Lickorish, gives a complete solution of problem 13 of (1).

V. P. Snaith has shown that a number of interesting homology theories (e.g. K-theory, complex cobordism) arise when one localizes the stable homotopy type of certain H-spaces (BU(1), BU, etc.). In this paper we sketch a construction which, given an H-space X and , produces a ring spectrum IΣ−nX which lies between Xand its localization with respect to b, and in many cases has a tractable structure. We apply this when X is an Eilenberg-MacLane space K(ℤ/, n) and is a generator (p = 0 or a prime), and get complete results on the localizations in this case. The results confirm one's suspicions that localization is a fairly drastic manoeuvre, cutting down the large and complicated ring to something more or less trivial, except when p = 0 and n = 2, in which case the rich geometry of K(ℤ, 2) = BU(1) gives complex K-theory.

Classical bifurcation theorems for a 1 -parameter family of plane dynamical systems

assert the presence of closed orbits clustering at some distinguished parameter value (∈ = 0, say). Here, for any ∈, the origin is the only stationary point. The topological content of the mostly analytic hypotheses imposed is some change in the stability behaviour of the origin at ∈ = 0, roughly the passing of a kind of stability to a kind of instability. Topologically speaking, e.g. some of the conditions demanded are asymptotic stability of the origin for the negative system at ∈ > 0 and asymptotic stability of the origin for at ∈ < 0 (Hopf (8), Ruelle and Takens(11)) or ∈ = 0 (Chafee(2)).

Fall, Arveson and Muhly(4) characterized the compact perturbation of nest algebras. In fact they proved that the compact perturbation of a nest algebra corresponding to a nest of projections is the algebra of operators which are quasitriangular relative to this nest. Erdos and Power(3) investigated weakly closed ideals and modules of nest algebras and these exhibit properties that are very close to the properties of the nest algebras themselves. They also showed that in certain cases, as in the case when the homomorphism which determines the nest algebra module is continuous, the results of Fall, Arveson and Muhly carry over to the more general situation. In this paper we provide a characterization of the compact perturbation of any nest algebra module.

A JB-algebra is a real Jordan algebra A which is also a Banach space and whose norm and multiplication satisfy the two following conditions

(i) ∥a2∥ = ∥a∥2,

(ii) ∥a2 − b2∥ ≤ max{∥a2∥, ∥b2∥},

for all elements a and b in A. A JB-algebra which is also a Banach dual space is called a JBW-algebra. The properties of JB-algebras and JBW-algebras can be found in (3), (4), (8) and (15).

In (2), indefinite Kählerian manifolds have been examined from the point of view of holomorphic sectional curvature. Examples in (2) show that the analogue of Kulkarni's theorem (see (4), p. 173) for the holomorphic sectional curvature is false and the best possible result in the direction is:

Theorem 1 (known, (2)). Let M be a connected indefinite Kählerian manifold with complex dimension n ≥ 2. If the holomorphic sectional curvature of M is bounded above and bounded below, then M is an indefinite complex space form.

If Z and E are Hausdorff locally convex spaces (LCS) then by Lb(Z, E) we mean the space of continuous linear maps from Z to E endowed with the topology of uniform convergence on the bounded subsets of Z. The dual Lb(Z, E)′ will always carry the topology of uniform convergence on the bounded subsets of Lb(Z, E). If K(Z, E) is a linear subspace of L(Z, E) then Kb(Z, E) will be used to denote K(Z, E) with the relative topology and Kb(Z, E)″ will mean the dual of Kb(Z, E)′ with the natural topology of uniform convergence on the equicontinuous subsets of Kb(Z, E)′. If Z and E are Banach spaces these provide, in each instance, the usual norm topologies.

A real Banach space (X, ‖ · ‖) is said to be uniformly convex (UC) (or uniformly rotund) if for all ∈ > 0 there is a δ > 0 such that if ‖x| ≤ 1, ‖y‖ ≤ 1 and ‖x−y‖ ≥ ∈, then ‖(x + y)/2‖ ≤ 1− δ.

Let G be a metrizable non-discrete locally compact group. Let M(G) be the convolution measure algebra of G. We shall denote by Lx and Rx, x β G, the left and the right translation operation on M(G), respectively. A measure μ β M(G) has strongly independent powers or μ is a s.i.p. measure, if, for x β G and n, m β N,

whenever n ≠ m or x ≠ e, where e is the identity of G. If a measure μ satisfies (1) for x = e and n + m, we say that μ has independent powers, or μ is an i.p. measure. Notice that the powers μn and μm of μ are convolution powers.

The following interesting result on strongly summable sequences is due to Kuttner (2).

Theorem 1. Let 0 < p < 1. Then given any Toeplitz matrix A there is a sequence summable wp which is not summable A.

As usual, for 0 < p < ∞, we denote by wp the space of strongly summable sequences with index p. Thus x = (xk) ∈ wp if and only if there is a number l such that

Strongly summable sequences arise in the theory of Fourier series, see e.g. (8), Chapter X, in Ergodic theory, see e.g. (1), as well as in summability theory.