Let $p$ be a prime and $\vartheta$ an integer of order $t$ in the multiplicative group modulo $p$. In this paper, we continue the study of the distribution of Diffie–Hellman triples$(\vartheta^x, \vartheta^y, \vartheta^{xy})$ by considering the closely related problem of estimating exponential sums formed from linear combinations of the entries in such triples. We show that the techniques developed earlier for complete sums can be combined, modified and developed further to treat incomplete sums as well. Our bounds imply uniformity of distribution results for Diffie–Hellman triples as the pair $(x,y)$ varies over small boxes.

In this paper we consider the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. Using new exponential sums in tandem with a sieve method we are able to provide stronger “minor arc” estimates than previous authors, thereby improving the saving obtained in the exponent by a factor 8/7.

We define a group theoretic criterion on a pro-$p$ group that guarantees the existence of an analytic structure over ${\mathbb Z}_p[[t]]$. This is a one-way analogue of Lazard's succesful answer to the $p$-adic version of Hilbert's Fifth Problem. In the course of the proof we define a $T$-map, a $T$-uniform group and set up a category equivalence between $T$-uniform pro-$p$ groups and powerful ${\mathbb Z}_p[[t]]$-Lie algebras.

Assume that all spaces and maps are localised at a fixed prime $p$. We study the possibility of generating a universal space $U(X)$ from a space $X$ which is universal in the category of homotopy associative, homotopy commutative $H$-spaces in the sense that any map $f\colon X\to Y$ to a homotopy associative, homotopy commutative $H$-space extends to a uniquely determined $H$-map $\overline{f}\colon U(X)\to Y$. Developing a method for recognising certain universal spaces, we show the existence of the universal space $F_2(n)$ of a certain three-cell complex $L$. Using this specific example, we derive some consequences for the calculation of the unstable homotopy groups of spheres, namely, we obtain a formula for the $d_1$-differential of the EHP-spectral sequence valid in a certain range.

Let $F$ be a free group, freely generated by a non-empty set $X$ and let $R$ be a cyclically reduced word in $F$. Let $X_0$ be the subset of elements of $X$ which occur in $R$ or in $R^{-1}$. Let $G$ be the quotient of $F$ by the normal closure of $R$. By the classical Dehn–Magnus Freiheitssatz, if $Y$ is a subset of $X$ which does not contain $X_0$, then the subgroup of $G$ generated by the image of $Y$ in $G$ is freely generated by it.

Free products of groups can be viewed as generalisations of free groups, by replacing the infinite cyclic groups generated by individual elements of $X$, by arbitrary groups. It is known that in special cases the corresponding version of the Freiheitssatz holds true for one-relator quotients of free products of two groups (components), for example if the relator satisfies a small cancellation condition.

In this work we extend Magnus' Freiheitssatz to an arbitrary number of components and where the subset corresponding to $Y$ above contains arbitrary number of components, provided that the relator satisfies the small cancellation condition C(8) and some further conditions. As an application we imitate, under certain conditions, the result of Magnus stating that one-relator groups have the structure of HNN extensions.

The method of proof includes a combinatorial analysis of van Kampen diagrams, relying on the structure theorem for small cancellation diagrams.

Let $(R, {\frak m})$ be a local Cohen–Macaulay ring and let $I$ be an ${\frak m}$-primary ideal. In this paper we give sharp bounds on the multiplicity of the special fiber ring ${\mathcal F}$ of $I$ in terms of other well-known invariants of $I$. Special attention is then paid in studying when equality holds in these bounds, with a particular interest in the unmixedness or, better, the Cohen–Macaulayness of ${\mathcal F}$.

We define torsors for sheaves of simplicial groups and sheaves of groupoids enriched in simplicial sets, and give classification results for these torsors in terms of the homotopy theory of simplicial sheaves. The proofs of the classification results use a new, general approach to cocycles taking values in simplicial sheaves. We prove a homotopy classification result for gerbes locally isomorphic to a fixed sheaf of groups.

Functional analytic properties of the Banach Algebras $A_p^r(G)$ such as Arens regularity, amenability, weak sequential completeness, sets of synthesis and strict containment for fixed $p$ and increasing $r$, are investigated.

It is found that $A_p^r(G)$, $1\leq r<\infty$, behave very differently to the Figa–Talamanca–Herz algebras $A_p(G)$.

Let $\varphi\,{:}\,G \,{\rightarrow}\,{\rm O}(d)$ be an orthogonal representation of a group $G$. A finite subset $X$ of a Euclidean space ${\bf R}^d$ is said to be subtransitive under$(G, \varphi)$ when it is a subset of an orbit under the action of ${\varphi }(G)$; furthermore, we say that $X$ is finitely subtransitive when it is subtransitive under $(G, \varphi)$ for some finite group $G$. If $X$ is subtransitive under a representation $\varphi\,{:}\,G\,{\rightarrow}\, {\rm O}(d)$ then it is clear that $X$ is spherical; that is, contained in a sphere in the representation space ${\bf R}^d$. In connection with Euclidean Ramsey theory, I.B. Leader [6] has posed the converse question:

Is is true that every finite spherical set is finitely subtransitive?

The paper deals with analytic families of planar vector fields, studying methods to detect the cyclicity of a non-isolated closed orbit, i.e. the maximum number of limit cycles that can locally bifurcate from it. It is known that this multi-parameter problem can be reduced to a single-parameter one, in the sense that there exist analytic curves in parameter space along which the maximal cyclicity can be attained. In that case one speaks about a maximal cyclicity curve (mcc) in case only the number is considered and of a maximal multiplicity curve (mmc) in case the multiplicity is also taken into account. In view of obtaining efficient algorithms for detecting the cyclicity, we investigate whether such mcc or mmc can be algebraic or even linear depending on certain general properties of the families or of their associated Bautin ideal. In any case by well chosen examples we show that prudence is appropriate.

Recent computations of UNil-groups by Connolly, Ranicki and Davis are used to study splittability of homotopy equivalences between 4-dimensional manifolds with infinite dihedral fundamental groups.

In 1977 Grünbaum and Shephard described all possible 93 types of isohedral marked tilings of the plane; 46 of them are called basic, since their induced tile group is trivial. The aim of this paper is to give an algebraic description of all basic tilings. A purely algebraic characterization of the adjacency symmetries of tiles of the 46 basic tilings is presented. Moreover, 46 related abstract definitions of two-dimensional crystallographic groups supplement and extend those of the well-known book Generators and Relations for Discrete Groups by Coxeter and Moser.

Let $p$ be an odd prime and $G$ the finite cyclic group of order $p$. We use the Casson–Walker–Lescop invariant to find a necessary condition for a three-manifold to have an action of $G$ with a circle as the set of fixed points and an integral homology sphere as the quotient.

It is well known that every orientation preserving homeomorphism of the plane $\mathbb{R}^2$ with a $k$-periodic orbit ${\cal O}$ ($k \geq 2$) necessarily has a fixed point (see [4] or [5], [7], [11]). It is then interesting to know if ${\cal O}$ is linked with one of these fixed points. The difficulty in answering this question depends a lot on the precise sense we give to the word “linked”. Before discussing briefly this point, let us observe this general problem has a very natural counterpart in the framework of orientation reversing homeomorphisms of the sphere $\mathbb{S}^2$. Indeed, every such homeomorphism possessing a $k$-periodic orbit ${\cal O}$ ($k\geq 3$) also has a 2-periodic orbit ([2]) and one can ask if ${\cal O}$ is linked with one of these 2-periodic orbits. The aim of this paper is to try to answer this question.

Let $E(1)_K$ denote the homotopy rational elliptic surface corresponding to a knot $K$ in $S^3$ constructed by R. Fintushel and R. J. Stern. We construct an infinite family of homologous non-isotopic symplectic tori representing a primitive 2-dimensional homology class in $E(1)_K$ when $K$ is any nontrivial fibred knot in $S^3$. We also show how these tori can be non-isotopically embedded as homologous symplectic submanifolds in other symplectic 4-manifolds.

Arithmetic and non arithmetic hyperbolic 3-manifolds having infinitely many fibrations over $S^{1}$ of arbitrary high genus are constructed. A concrete example is studied in detail.

Topological surgery in dimension four is used to show a two component link with Alexander polynomial one is topologically concordant to the Hopf link.

We investigate the behavior of the higher-order degrees, $\bar{\delta}_n$, of a finitely presented group $G$. These $\bar{\delta}_n$ are functions from $H^1(G;{\mathbb Z})$ to ${\mathbb Z}$ whose values are the degrees certain higher-order Alexander polynomials. We show that if ${\rm def}(G)\geq 1$ or $G$ is the fundamental group of a compact, orientable 3-manifold then $\bar{\delta}_n$ is a monotonically increasing function of $n$ for $n\geq1$. This is false for general groups. As a consequence, we show that if a 4-manifold of the form $X \times S^1$ admits a symplectic structure then X “looks algebraically like” a 3-manifold that fibers over $S^1$, supporting a positive answer to a question of Taubes. This generalizes a theorem of S. Vidussi and is an improvement on previous results of the author. We also find new conditions on a 3-manifold $X$ that will guarantee that the Thurston norm of $f^{\ast}(\psi)$, for $\psi \in H^1(X;{\mathbb Z})$ and $f{:}\,Y \rightarrow X$ a surjective map on $\pi_1$, will be at least as large the Thurston norm of $\psi$. When $X$ and $Y$ are knot complements, this gives a partial answer to a question of J. Simon.

More generally, we define $\Gamma$-degrees, $\bar{\delta}_\Gamma$, corresponding to a surjective map $G\twoheadrightarrow \Gamma$ for which $\Gamma$ is poly-torsion-free-abelian. Under certain conditions, we show they satisfy a monotonicity condition if one varies the group $\Gamma$. As a result, we show that these generalized degrees give obstructions to the deficiency of a group being positive and obstructions to a finitely presented group being the fundamental group of a compact, orientable 3-manifold.

Using convex surfaces and Kanda's classification theorem, we classify Legendrian isotopy classes of Legendrian linear curves in all tight contact structures on $T^3$. Some of the knot types considered in this paper provide new examples of non transversally simple knot types.

We study different localization techniques both for classical and motivic spectra. In particular, we show that the motivic model structures of Morel and Voevodsky are cellular in the sense of Hirschhorn and thus may be further localized. As a possible application, we discuss a motivic version of the chromatic filtration and the chromatic spectral sequence.