Si un réseau G de Rn contient Zn et est permis pour l'«octaèdre généralisé» Ωn d'équation

l'indice de Zn dans G est borné supérieurement par n! d'après les théorèmes généraux de Minkowski.

In 1961, S. T. Hu published a paper (1) in which he discussed the desirability of discovering those topological properties which are preserved under homotopy and isotopy equivalences. In that paper he gave general tests in terms of weakly hereditary and hereditary topological properties for homotopy and isotopy properties.

In this paper, general tests for homotopy and isotopy properties in terms of weakly hereditary properties and of a class of properties which the author calls open properties are given. In the last sections, we shall show the strong role played by the notions of dimension and separating subsets in forming isotopy properties.

In [3], the study of (p;q) radicals was initiated. In this paper, the integral polynomials p(x) and q(x) which determine the Jacobson radical are characterized and the Jacobson radical is shown to be the only semiprime (p;q) radical for which all fields are semisimple. Also, it is observed that the prime, nil, and Brown-McCoy radicals are not (p;q) radicals. To show that the semiprime (p;q) radicals are special and that they can be determined by subclasses of the class of primitive rings, a classification theorem for (p;q)-regular primitive rings is given. Finally, it is shown that the collection of semiprime (p;q) radicals and the collection of semiprime (p;1) radicals coincide.

If G = (G, τ) is a topological group with topology τ, then there is a smallest topology τ* ⊇ τ such that G* = (G, τ*) is a topological group with equal left and right uniformities (1). Bagley and Wu introduced this topology in (1), and studied the relationship between Gand G*. In this paper we prove some additional results concerning G* and groups with equal uniformities in general. The structure of locally compact groups with equal uniformities has been studied extensively. If G is a locally compact connected group, then G has equal uniformities if and only if G ≅ V× K,where F is a vector group and Kis a compact group (5). More generally, every locally compact group with equal uniformities has an open normal subgroup of the form V× K(4).

We shall show that if a Riemann surface is continuable, then it admits one of three types of continuations. Using this classification of continuations, we construct two nontrivial examples of two-sheeted unlimited covering Riemann surfaces of the unit disk one of which is continuable and the other is not.

For $p$ a prime, a $p$-typical cover of a connected scheme on which $p\,=\,0$ is a finite étale cover whose monodromy group (i.e.,the Galois group of its normal closure) is a $p$-group. The geometry of such covers exhibits some unexpectedly pleasant behaviors; building on work of Katz, we demonstrate some of these. These include a criterion for when a morphism induces an isomorphism of the $p$-typical quotients of the étale fundamental groups, and a decomposition theorem for $p$-typical covers of polynomial rings over an algebraically closed field.

In group theory Schreier's technique provides a basis for a subgroup of a free group. In this paper an analogue is developed for free Lie algebras. It hinges on the idea of cutting a Hall set into two parts. Using it, we show that proper subalgebras of finite codimension are not finitely generated and, following M. Hall, that a finitely generated subalgebra is a free factor of a subalgebra of finite codimension.

In this paper, we characterize unitary representations of $\pi \text{ }:=\text{ }\pi {{\text{ }}_{1}}({{S}^{2}}\backslash \{{{s}_{1}},\ldots ,{{s}_{l}}\})$ whose generators ${{u}_{1}},\,\ldots ,\,{{u}_{l}}$ (lying in conjugacy classes fixed initially) can be decomposed as products of two Lagrangian involutions ${{u}_{j}}\,=\,{{\sigma }_{j}}{{\sigma }_{j+1}}$ with ${{\sigma }_{l+1}}\,=\,{{\sigma }_{1}}$. Our main result is that such representations are exactly the elements of the fixed-point set of an anti-symplectic involution defined on the moduli space ${{\mathcal{M}}_{C}}\,:\,=\,\text{Ho}{{\text{m}}_{C}}(\pi ,\,U(n))\,/\,U(n)$. Consequently, as this fixed-point set is non-empty, it is a Lagrangian submanifold of ${{\mathcal{M}}_{C}}$. To prove this, we use the quasi-Hamiltonian description of the symplectic structure of ${{\mathcal{M}}_{C}}$ and give conditions on an involution defined on a quasi-Hamiltonian $U$-space $(M,\,\omega ,\,\mu :\,M\to \,U)$ for it to induce an anti-symplectic involution on the reduced space $M//U:=\,{{\mu }^{-1}}\,(\{1\})/U$.

S. K. Sehgal ([9], Problem 26) proposed the following question : Let A, B be rings and X an infinite cyclic group. Does AX ⋍ BX imply A ⋍ B? M. M. Parmenter and S. K. Sehgal (cf. [9], Chapter 4) proved that, under some strong assumptions concerning rings A, B, the answer is affirmative. In this paper, we show that the assumptions concerning the ring B may be omitted in the above mentioned results. Moreover, it is proven that if (AX)X ⋍ BX then AX ⋍ B for all rings A, B. If A is commutative and noetherian then a partial answer to Problem 27, [9] follows from our results.

Recently, L. Griinenfelder and M. M. Parmenter constructed nonisomorphic rings A, B for which the group rings AX, BX are isomorphic, [2], We give a new class of rings of this type. Our examples are noncommutative domains and algebras over finite fields. That also gives a negative answer to Problem 29, [9].

Recently, a great deal of attention has been paid to the concept of quasipure injectivity introduced by L. Fuchs as Problem 17 in [5]. An abelian group G is said to be quasi-pure-injective (q.p.i.) if every homomorphism from a pure subgroup of G to G can be lifted to an endomorphism of G. D. M. Arnold, B. O'Brien and J. D. Reid have succeeded in [1] to characterize torsion free q.p.i. of finite rank, whereas in [2] we solved the torsion case and in [3] we studied certain classes of infinite rank torsion free q.p.i. groups.

Let $\beta:\,{{S}^{2n+1}}\,\to \,{{S}^{2n+1}}$ be a minimal homeomorphism $\left( n\,\ge \,1 \right)$. We show that the crossed product $C\left( {{S}^{2n+1}} \right)\,{{\rtimes }_{\beta}}\mathbb{Z}$ has rational tracial rank at most one. Let $\Omega $ be a connected, compact, metric space with finite covering dimension and with ${{H}^{1}}\left( \Omega ,\,\mathbb{Z} \right)\,=\,\left\{ 0 \right\}$. Suppose that ${{K}_{i}}\left( C\left( \Omega \right) \right)\,=\,\mathbb{Z}\,\oplus \,{{G}_{i}}$, where ${{G}_{i}}$ is a finite abelian group, $i\,=\,0,\,1$. Let $\beta :\,\Omega \,\to \,\Omega $ be a minimal homeomorphism. We also show that $A\,=\,C\left( \Omega \right)\,{{\rtimes }_{\beta}}\,\mathbb{Z}$ has rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This is done by studying minimal homeomorphisms on $X\,\times \,\Omega $, where $X$ is the Cantor set.

If the Lp spaces of two measure spaces are “the same”, to what extent can we identify the underlying measure spaces? This question has been partially answered by Schneider [7] (extending results of Forelli [2]). He proves that a linear isometry between the Lv spaces of two finite measure spaces is in fact an (isometric) homomorphism between the corresponding L∞ spaces, if it preserves the identity.

In previous papers, Barr and Raphael investigated the situation of a topological space $Y$ and a subspace $X$ such that the induced map $C(Y)\,\to \,C(X)$ is an epimorphism in the category $\mathcal{C}\mathcal{R}$ of commutative rings (with units). We call such an embedding a $\mathcal{C}\mathcal{R}$-epic embedding and we say that $X$ is absolute $\mathcal{C}\mathcal{R}$-epic if every embedding of $X$ is $\mathcal{C}\mathcal{R}$-epic. We continue this investigation. Our most notable result shows that a Lindelöf space $X$ is absolute $\mathcal{C}\mathcal{R}$-epic if a countable intersection of $\beta X$-neighbourhoods of $X$ is a $\beta X$-neighbourhood of $X$. This condition is stable under countable sums, the formation of closed subspaces, cozero-subspaces, and being the domain or codomain of a perfect map. A strengthening of the Lindelöf property leads to a new class with the same closure properties that is also closed under finite products. Moreover, all $\sigma $-compact spaces and all Lindelöf $P$-spaces satisfy this stronger condition. We get some results in the non-Lindelöf case that are sufficient to show that the Dieudonné plank and some closely related spaces are absolute $\mathcal{C}\mathcal{R}$-epic.

We prove that two, apparently different, class-group valued Galois module structure invariants associated to the algebraic $K$-groups of rings of algebraic integers coincide. This comparison result is particularly important in making explicit calculations.

Let be a semigroup of measure preserving transformations on a measure space (Ω, ℱ, μ). The main result of the paper is the proof of a.e. convergence for the moving averages where {FIn} is a superadditive process and {In} is a sequence of cubes in satisfying the "cone-condition". The identification of the limit is given. A moving local theorem is also proved.

In [7; 8; 11] it is shown that an appropriate map of a planar surface into a 3-manifold can be replaced by an embedding. In [1 ; 4 ; 6 ; 7 ; 9 ; 10] conditions are given so that a "non-trivial" map of a planar surface (2-sphere) can be replaced by a non-trivial embedding of a planar surface (2-sphere). In this paper we give conditions on an annular map which guarantee the existence of a non-trivial embedding of an annulus. It is reported that F. Waldhausen has proved a similar but stronger "annulus theorem".

This note concerns itself primarily with the representation of continuous multiplicative functionals on L2 types of rings or Banach algebras to the real or complex fields where convolution is taken as the ring multiplication. In a recent publication [1] such functionals were studied for the continuous function ring C(S) over a compact space S.

It was proved almost forty years ago that every mapping of a tree into itself has at least one fixed point, but not much is known so far about the structure of the possible fixed point sets. One topic related to this question, the study of homeomorphisms and monotone mappings of trees which leave an end point fixed, was first considered by G. E. Schweigert [6] and continued by L. E. Ward, Jr. [8] and others. One result by Schweigert and Ward is the following: any monotone mapping of a tree onto itself which leaves an end point fixed, also leaves at least one other point fixed.

It is further known that not only single-valued mappings, but also upper semi-continuous (use) and connected-valued multifunctions of trees have a fixed point [7], and that two use and biconnected multifunctions from one tree onto another have a coincidence [5].

The work of various authors (e.g. Frink [3] and Markushevitch [7]) suggests the possibility of studying complete biorthogonal sequences in Banach spaces as a generalization of orthogonal families of continuous functions. But except for the case where the complete biorthogonal sequence is a Schauder basis such studies have not led to a very rich theory. The main reason for this is that an arbitrary complete biorthogonal sequence is not likely to have many helpful properties. For instance, in every separable Banach space X one can find a complete biorthogonal sequence {ei, Ei} which is not one-summable.

Various characteristic conjugacy classes of subgroups having covering/avoidance properties with respect to chief factors have recently played a major role in the study of finite soluble groups. Apart from the subgroups which are now called Hall subgroups, P. Hall [7] also considered the system normalizers of a finite soluble group and showed that these form a characteristic conjugacy class, cover the central chief factors and avoid the rest. The system normalizers were later shown by Carter and Hawkes [1] to be the simplest example of a wealth of characteristic conjugacy classes of subgroups of finite soluble groups which arise naturally as a consequence of the theory of formations.