Cohomological dimension and metrizable spaces. II

The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : . Theorem. Suppose are subsets of a metrizable space and and are CW complexes. If is an absolute extensor for and is an absolute extensor for , then the join is an absolute extensor for .

As an application we prove the following analogue of the Menger-Urysohn Theorem for cohomological dimension: Theorem. Suppose are subsets of a metrizable space. Then

for any ring with unity and

for any abelian group .

The second part of the paper is devoted to the question of existence of universal spaces: Theorem. Suppose is a sequence of CW complexes homotopy dominated by finite CW complexes. Then

a.

Given a separable, metrizable space such that , , there exists a metrizable compactification of such that , .

b.

There is a universal space of the class of all compact metrizable spaces such that for all .

c.

There is a completely metrizable and separable space such that for all with the property that any completely metrizable and separable space with for all embeds in as a closed subset.

     

Whitehead test modules

A (right -) module is said to be a Whitehead test module for projectivity (shortly: a p-test module) provided for each module , implies is projective. Dually, i-test modules are defined. For example, is a p-test abelian group iff each Whitehead group is free. Our first main result says that if is a right hereditary non-right perfect ring, then the existence of p-test modules is independent of ZFC + GCH. On the other hand, for any ring , there is a proper class of i-test modules. Dually, there is a proper class of p-test modules over any right perfect ring.

A non-semisimple ring is said to be fully saturated (-saturated) provided that all non-projective (-generated non-projective) modules are i-test. We show that classification of saturated rings can be reduced to the indecomposable ones. Indecomposable 1-saturated rings fall into two classes: type I, where all simple modules are isomorphic, and type II, the others. Our second main result gives a complete characterization of rings of type II as certain generalized upper triangular matrix rings, . The four parameters involved here are skew-fields and , and natural numbers . For rings of type I, we have several partial results: e.g. using a generalization of Bongartz Lemma, we show that it is consistent that each fully saturated ring of type I is a full matrix ring over a local quasi-Frobenius ring. In several recent papers, our results have been applied to Tilting Theory and to the Theory of -modules.

     

Abstract functions with continuous differences and Namioka spaces

Let be a semigroup and a topological space. Let be an Abelian topological group. The right differences of a function are defined by for . Let be continuous at the identity of for all in a neighbourhood of . We give conditions on or range under which is continuous for any topological space . We also seek conditions on under which we conclude that is continuous at for arbitrary . This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.

     

On the ordering of -modal cycles

The forcing relation on -modal cycles is studied. If is an -modal cycle then the -modal cycles with block structure that force form a -horseshoe above . If -modal forces , and does not have a block structure over , then forces a -horseshoe of simple extensions of .

     

Powers in Finitely Generated Groups

In this paper we study the set of -powers in certain finitely generated groups . We show that, if is soluble or linear, and contains a finite index subgroup, then is nilpotent-by-finite. We also show that, if is linear and has finite index (i.e. may be covered by finitely many translations of ), then is soluble-by-finite. The proof applies invariant measures on amenable groups, number-theoretic results concerning the -unit equation, the theory of algebraic groups and strong approximation results for linear groups in arbitrary characteristic.

     

Distinguished representations and quadratic base change for

Let be a quadratic extension of number fields. Suppose that every real place of splits in and let be the unitary group in 3 variables. Suppose that is an automorphic cuspidal representation of . We prove that there is a form in the space of such that the integral of over is non zero. Our proof is based on earlier results and the notion, discussed in this paper, of Shalika germs for certain Kloosterman integrals.

     

Nonnegative Radix Representations for the Orthant

Let be a nonnegative real matrix which is expanding, i.e. with all eigenvalues , and suppose that is an integer. Let consist of exactly nonnegative vectors in . We classify all pairs such that every in the orthant has at least one radix expansion in base using digits in . The matrix must be a diagonal matrix times a permutation matrix. In addition must be similar to an integer matrix, but need not be an integer matrix. In all cases the digit set can be diagonally scaled to lie in . The proofs generalize a method of Odlyzko, previously used to classify the one--dimensional case.

     

Regularity and Algebras of Analytic Functions in Infinite Dimensions

A Banach space is known to be Arens regular if every continuous linear mapping from to is weakly compact. Let be an open subset of , and let denote the algebra of analytic functions on which are bounded on bounded subsets of lying at a positive distance from the boundary of We endow with the usual Fréchet topology. denotes the set of continuous homomorphisms . We study the relation between the Arens regularity of the space and the structure of .

     

A Concordance Extension Theorem

Let be a manifold approximate fibration between closed manifolds, where , and let be the mapping cylinder of . In this paper it is shown that if is any concordance on , then there exists a concordance such that and . As an application, if and are closed manifolds where is a locally flat submanifold of and and , then a concordance extends to a concordance on such that . This uses the fact that under these hypotheses there exists a manifold approximate fibration , where is a closed -manifold, such that the mapping cylinder is homeomorphic to a closed neighborhood of in by a homeomorphism which is the identity on .

     

Covering the integers by arithmetic sequences. II

Let ( be a system of arithmetic sequences where and . For system will be called an (exact) -cover of if every integer is covered by at least (exactly) times. In this paper we reveal further connections between the common differences in an (exact) -cover of and Egyptian fractions. Here are some typical results for those -covers of : (a) For any there are at least positive integers in the form where . (b) When (, either or , and for each positive integer the binomial coefficient can be written as the sum of some denominators of the rationals if forms an exact -cover of . (c) If is not an -cover of , then have at least distinct fractional parts and for each there exist such that (mod 1). If forms an exact -cover of with or () then for every and there is an such that (mod 1).

     

The Bergman kernel function of some Reinhardt domains

The boundary behavior of the Bergman Kernel function of some Reinhardt domains is studied. Upper and lower bounds for the Bergman kernel function are found at the diagonal points . Let be the Reinhardt domain

where , ; and let be the Bergman kernel function of . Then there exist two positive constants and and a function such that

holds for every . Here

and is the defining function for . The constants and depend only on and , not on .

     

On quadratic forms of height two and a theorem of Wadsworth

Let and be anisotropic quadratic forms over a field of characteristic not . Their function fields and are said to be equivalent (over ) if and are isotropic. We consider the case where and is divisible by an -fold Pfister form. We determine those forms for which becomes isotropic over if , and provide partial results for . These results imply that if and are equivalent and , then is similar to over . This together with already known results yields that if is of height and degree or , and if , then and are equivalent iff and are isomorphic over .

     

Uniform harmonic approximation of bounded functions

Let be an open set in and be a relatively closed subset of . We characterize those pairs which have the following property: every function which is bounded and continuous on and harmonic on can be uniformly approximated by functions harmonic on . Several related results concerning both harmonic and superharmonic approximation are also established.

     

Functorial structure of units in a tensor product

The behavior of units in a tensor product of rings is studied, as one factor varies. For example, let be an algebraically closed field. Let and be reduced rings containing , having connected spectra. Let be a unit. Then for some units and .

Here is a deeper consequence, stated for simplicity in the affine case only. Let be a field, and let be a homomorphism of finitely generated -algebras such that is dominant. Assume that every irreducible component of or is geometrically integral and has a rational point. Let be a faithfully flat homomorphism of reduced -algebras. For a -algebra, define to be . Then satisfies the following sheaf property: the sequence

is exact. This and another result are used to prove (5.2) of [7].

     

On extension of cocycles to normalizer elements, outer conjugacy, and related problems

Let be an ergodic automorphism of a Lebesgue space and a cocycle of with values in an Abelian locally compact group . An automorphism from the normalizer of the full group is said to be compatible with if there is a measurable function such that at a.e. . The topology on the set of all automorphisms compatible with is introduced in such a way that becomes a Polish group. A complete system of invariants for the -outer conjugacy (i.e. the conjugacy in the quotient group is found. Structure of the cocycles compatible with every element of is described.

     

A group of paths in

We define a group structure on the set of compact ``minimal' paths in . We classify all finitely generated subgroups of this group : they are free products of free abelian groups and surface groups. Moreover, each such group occurs in . The subgroups of isomorphic to surface groups arise from certain topological -forms on the corresponding surfaces. We construct examples of such -forms for cohomology classes corresponding to certain eigenvectors for the action on cohomology of a pseudo-Anosov diffeomorphism. Using we construct a non-polygonal tiling problem in , that is, a finite set of tiles whose corresponding tilings are not equivalent to those of any set of polygonal tiles. The group has applications to combinatorial tiling problems of the type: given a set of tiles and a region , can be tiled by translated copies of tiles in ?

     

Topological centers of certain dual algebras

Let be a Banach algebra with a bounded approximate identity. Let and be, respectively, the topological centers of the algebras and . In this paper, for weakly sequentially complete Banach algebras, in particular for the group and Fourier algebras and , we study the sets , , the relations between them and with several other subspaces of or .

     

On vanishing of characteristic numbers in Poincaré complexes

Let be the evaluation subgroup as defined by Gottlieb. Assume the Hurewicz map is non-trivial and is a field. We will prove: if is a Poincaré complex oriented in -coefficient, all the characteristic numbers of in -coefficient vanish. Similarly, if and is a -Poincaré complex, then all the mod Wu numbers vanish. We will also show that the existence of a non-trivial derivation on with some suitable conditions implies vanishing of mod Wu numbers.

     

A Groenewold-Van Hove Theorem for

We prove that there does not exist a nontrivial quantization of the Poisson algebra of the symplectic manifold which is irreducible on the su(2) subalgebra generated by the components of the spin vector. In fact there does not exist such a quantization of the Poisson subalgebra consisting of polynomials in . Furthermore, we show that the maximal Poisson subalgebra of containing that can be so quantized is just that generated by .

     

Tensor products over abelian Walgebras

Tensor products of Calgebras over an abelian Walgebra are studied. The minimal Cnorm on is shown to be just the quotient of the minimal Cnorm on if or is exact.