In this paper we study the symplectic relations appearing as the generalized cotangent bundle liftings of smooth stable mappings. Using this class of symplectic relations the classification theorem for generic (pre) images of lagrangian submanifolds is proved. The normal forms for the respective classified puilbacks and pushforwards are provided and the connections between the singularity types of symplectic relation, mapped lagrangian submanifold and singular image, are established. The notion of special symplectic triplet is introduced and the generic local models of such triplets are studied. We show that the open swallowtails are canonically represented as pushforwards of the appropriate regular lagrangian submanifolds. Using the SL2(ℝ) invariant symplectic structure of the space of binary forms of n appropriate dimension we derive the generating families for the open swallowtails and the respective generating functions for its regular resolutions.

The discovery by V. F. R. Jones of a Laurent polynomial invariant VL(t)∈ℤ[t±½] for every oriented link L in the 3-sphere prompted the finding of a more general 2-variable Laurent polynomial invariant PL(l, m)∈ℤ[l±1,m±1], (see [4], [3] and [6]).

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.

In an earlier paper [10], we constructed a ‘locally finite approximation away from a given prime p’ of the classifying space BG of a Lie group with finite component group. Such an approximation consists of a locally finite group g and a homotopy class of maps which in particular induces an isomorphism in cohomology with finite coefficients of order prime to p. The usefulness of such a construction is that it reduces various homotopy-theoretic questions concerning the space BG to the corresponding questions concerning Bπ for finite subgroups π. For example, we demonstrated in [10] how H. Miller's proof of the Sullivan conjecture concerning maps from , where π is a finite group and X is a finite-dimensional complex, can be extended to maps BG→X for G a Lie group with finite component group.

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.

Elementary observations yield new classes of knotted 2-spheres in S4 which do not admit Punct (# S1 × S2) as a Seifert manifold. This provides a rather painless proof which re-establishes the existence of non-ribbon 2-knots.

Let S+ (resp. S−) denote the class of fundamental groups of closed orientable (resp. non-orientable) 2-manifolds of genus ≥ 2, and let surface = S+ ∪ S−. In the list of problems raised at the 1977 Durham Conference on Homological Group Theory occurs the following([7], p. 391, (G. 3)).

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.

Let M denote a compact topological 3-manifold, without boundary, foliated by topological circles in the sense of Seifert's gefaserter Räume[8]. This will be called a Seifert structure, or Seifert fibration on M; the leaves of the foliation are called (regular or exceptional) fibres. Our main result is the following theorem, reminiscent of the theorem of Borsuk—Ulam [1] stating that every continuous function from S3 to S2 takes at least one pair of antipodal points to the same value.

In this article we extend the study of ‘plumbing’ initiated in [10] to the case of fibred even-dimensional knots. Plumbing is a geometric operation on the fibre-surfaces of two fibred knots of the same dimension that produces another such knot.

A simple proof is given that if X is a normed vector lattice, 1 ≼ p ≼ ∞, and the norm in X is p-additive, then the norm is p-superadditive if p < ∞. If p = ∞, then X satisfies Kakutani's classical M-space condition. The proof uses duality but is otherwise elementary.

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.

We begin by giving several definitions. A knot K in S3 is said to be amphicheiral if there is an orientation-reversing diffeomorphism h of S3 which leaves K setwise invariant. Suppose, in addition, that K is given an orientation. Then K is said to be positive amphicheiral if h preserves the orientation of K. If, in addition, the diffeomorphism h is an involution then K is strongly positive amphicheiral. Finally, we say a knot is slice if it bounds a smooth disc in B4. In this note we shall give a smooth example of a prime strongly positive amphicheiral knot which is not slice.

Sufficient conditions are given for a set to be a core for the generator of a weakly integrable semigroup on a locally convex space. The conditions are illustrated by semigroups of unbounded operators on a Banach space.

A spherical diagram over a 2-complex X consists of a tessellation T of the 2-sphere, together with a combinatorial map (that is, one which maps each cell homeomorphically on to a cell). In [2] and [6] a number of conjectures were made concerning spherical diagrams over a 2-complex X with H2(X) = 0. There are also related conjectures in [7]. The motivation for all of these conjectures is that they imply the Kervaire Conjecture: every non-singular system of equations over a group can be solved in some overgroup (see [1, 2, 4, 6, 7] for details and discussion).

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.

Let denote the Fourier series expansion of a function . Féver's celebrated theorem states that the series is summable (C, 1) at each point of continuity of f, where (C, 1) denotes the Cesàro method of summability of order 1. Riesz extended this result to (C, α) for each α > 0. Since then many authors have established sufficient conditions on various methods of summability to guarantee similar results. Over the years the pattern has been to strive for weaker conditions on the matrix, and to replace the condition of continuity on the function by a less stringent one. Theorems have been proved not only for the summability of f, but the summability of the derived series, and other series, related to f. Beginning with the work of Hille and Tamarkin[13], many of the theorems have been extended to absolute summability.

In [7] we gave a description of compact multipliers of Lω(X), and left open the question of whether weakly compact multipliers on Lω(X) are compact. This had already been answered for some examples of hypergroup algebras ([1], [2], [6] and [10]). Here we give a positive answer to this question in the general setting of a weighted hypergroup algebra.

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.

For a given locally convex space, it is always of interest to find conditions for its nuclearity. Well known results of this kind – by now already familiar – involve the use of tensor products, diametral dimension, bilinear forms, generalized sequence spaces and a host of other devices for the characterization of nuclear spaces (see [9]). However, it turns out, these nuclearity criteria are amenable to a particularly simple formulation in the setting of certain sequence spaces; an elegant example is provided by the so-called Grothendieck–Pietsch (GP, for short) criterion for nuclearity of a sequence space (in its normal topology) in terms of the summability of certain numerical sequences.