On various restricted sumsets

For finite subsets A_1,...,A_n of a field, their sumset is given by {a_1+...+a_n: a_1 in A_1,...,a_n in A_n}. In this paper we study various restricted sumsets of A_1,...,A_n with restrictions of the following forms: a_i-a_j not in S_{ij}, or alpha_ia_i not=alpha_ja_j, or a_i+b_i not=a_j+b_j (mod m_{ij}). Furthermore, we gain an insight into relations among recent results on this area obtained in quite different ways.Comment: 11 pages; final version for J. Number Theor

Acyclic Jacobi Diagrams

We propose a simple new combinatorial model to study spaces of acyclic Jacobi diagrams, in which they are identified with algebras of words modulo operations. This provides a starting point for a word-problem type combinatorial investigation of such spaces, and provides fresh insights on known results.Comment: 18 pages, 7 figures. Refernces added. Section 2 rewritten. Proof of Theorem 1.1 rewritten. To appear in Kobe J. Mat

A theorem of Hrushovski-Solecki-Vershik applied to uniform and coarse embeddings of the Urysohn metric space

A theorem proved by Hrushovski for graphs and extended by Solecki and Vershik (independently from each other) to metric spaces leads to a stronger version of ultrahomogeneity of the infinite random graph $R$, the universal Urysohn metric space \Ur, and other related objects. We show how the result can be used to average out uniform and coarse embeddings of \Ur (and its various counterparts) into normed spaces. Sometimes this leads to new embeddings of the same kind that are metric transforms and besides extend to affine representations of various isometry groups. As an application of this technique, we show that \Ur admits neither a uniform nor a coarse embedding into a uniformly convex Banach space.Comment: 23 pages, LaTeX 2e with Elsevier macros, a significant revision taking into account anonymous referee's comments, with the proof of the main result simplified and another long proof moved to the appendi

Stable concordance of knots in 3-manifolds

Knots and links in 3-manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato-Levine invariants, and Milnor's triple linking numbers. Besides fitting into a general theory of Whitney towers, these invariants provide obstructions to the existence of a singular concordance which can be homotoped to an embedding after stabilization by connected sums with $S^2\times S^2$. Results include classifications of stably slice links in orientable 3-manifolds, stable knot concordance in products of an orientable surface with the circle, and stable link concordance for many links of null-homotopic knots in orientable 3-manifolds.Comment: 59 pages, 28 figure

Higher order intersection numbers of 2-spheres in 4-manifolds

This is the beginning of an obstruction theory for deciding whether a map f:S^2 --> X^4 is homotopic to a topologically flat embedding, in the presence of fundamental group and in the absence of dual spheres. The first obstruction is Wall's self-intersection number mu(f) which tells the whole story in higher dimensions. Our second order obstruction tau(f) is defined if mu(f) vanishes and has formally very similar properties, except that it lies in a quotient of the group ring of two copies of pi_1(X) modulo S_3-symmetry (rather then just one copy modulo S_3-symmetry). It generalizes to the non-simply connected setting the Kervaire-Milnor invariant which corresponds to the Arf-invariant of knots in 3-space. We also give necessary and sufficient conditions for moving three maps f_1,f_2,f_3:S^2 --> X^4 to a position in which they have disjoint images. Again the obstruction lambda(f_1,f_2,f_3) generalizes Wall's intersection number lambda(f_1,f_2) which answers the same question for two spheres but is not sufficient (in dimension 4) for three spheres. In the same way as intersection numbers correspond to linking numbers in dimension 3, our new invariant corresponds to the Milnor invariant mu(1,2,3), generalizing the Matsumoto triple to the non simply-connected setting.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-1.abs.htm