Linear maps determining the norm topology

Let be a Banach function algebra on a compact space , and let be such that for any scalar the element is not a divisor of zero. We show that any complete norm topology on that makes the multiplication by continuous is automatically equivalent to the original norm topology of . Related results for general Banach spaces are also discussed.

     

Extension theory of separable metrizable spaces with applications to dimension theory

The paper deals with generalizing several theorems of the covering dimension theory to the extension theory of separable metrizable spaces. Here are some of the main results:

Generalized Eilenberg-Borsuk Theorem. Let be a countable CW complex. If is a separable metrizable space and is an absolute extensor of for some CW complex , then for any map , closed in , there is an extension of over an open set such that .

Theorem. Let be countable CW complexes. If is a separable metrizable space and is an absolute extensor of , then there is a subset of such that and .

Theorem. Suppose are countable, non-trivial, abelian groups and 0$">. For any separable metrizable space of finite dimension 0$">, there is a closed subset of with for .

Theorem. Suppose is a separable metrizable space of finite dimension and is a compactum of finite dimension. Then, for any , , there is a closed subset of such that and .

Theorem. Suppose is a metrizable space of finite dimension and is a compactum of finite dimension. If and are connected CW complexes, then

     

Tracially AF -algebras

Inspired by a paper of S. Popa and the classification theory of nuclear -algebras, we introduce a class of -algebras which we call tracially approximately finite dimensional (TAF). A TAF -algebra is not an AF-algebra in general, but a ``large' part of it can be approximated by finite dimensional subalgebras. We show that if a unital simple -algebra is TAF then it is quasidiagonal, and has real rank zero, stable rank one and weakly unperforated -group. All nuclear simple -algebras of real rank zero, stable rank one, with weakly unperforated -group classified so far by their -theoretical data are TAF. We provide examples of nonnuclear simple TAF -algebras. A sufficient condition for unital nuclear separable quasidiagonal -algebras to be TAF is also given. The main results include a characterization of simple rational AF-algebras. We show that a separable nuclear simple TAF -algebra satisfying the Universal Coefficient Theorem and having and is isomorphic to a simple AF-algebra with the same -theory.

     

surfaces of genus 8 and varieties of sums of powers of cubic fourfolds

The main result of this paper is that the variety of presentations of a general cubic form in variables as a sum of cubes is isomorphic to the Fano variety of lines of a cubic -fold , in general different from .

A general surface of genus determines uniquely a pair of cubic -folds: the apolar cubic and the dual Pfaffian cubic (or for simplicity and ). As Beauville and Donagi have shown, the Fano variety of lines on the cubic is isomorphic to the Hilbert scheme of length two subschemes of . The first main result of this paper is that parametrizes the variety of presentations of the cubic form , with , as a sum of cubes, which yields an isomorphism between and . Furthermore, we show that sets up a correspondence between and . The main result follows by a deformation argument.

     

A product formula for spherical representations of a group of automorphisms of a homogeneous tree, II

Let be the group of automorphisms of a homogeneous tree and let be the tensor product of two spherical irreducible unitary representations of . We complete the explicit decomposition of commenced in part I of this paper, by describing the discrete series representations of which appear as subrepresentations of .

     

A product formula for spherical representations of a group of automorphisms of a homogeneous tree, I

Let be the group of automorphisms of a homogeneous tree , and let be a lattice subgroup of . Let be the tensor product of two spherical irreducible unitary representations of . We give an explicit decomposition of the restriction of to . We also describe the spherical component of  explicitly, and this decomposition is interpreted as a multiplication formula for associated orthogonal polynomials.

     

Correction to ``Optimal factorization of Muckenhoupt weights'

Peter Jones' theorem on the factorization of  weights is sharpened for weights with bounds near , allowing the factorization to be performed continuously near the limiting, unweighted case. When and is an  weight with bound , it is shown that there exist  weights  such that both the formula and the estimates hold. The square root in these estimates is also proven to be the correct asymptotic power as .

     

Essential cohomology and extraspecial -groups

Let be an odd prime number and let be an extraspecial -group. The purpose of the paper is to show that has no non-zero essential mod- cohomology (and in fact that is Cohen-Macaulay) if and only if and .

     

Homotopy commutativity of -spaces with finitely generated cohomology

We show that a simply connected homotopy associative and homotopy commutative mod -space with finitely generated mod cohomology is homotopy equivalent to a finite product of , , the three-connected cover and the homotopy fiber of the map for . Our result also shows that a connected -space in the sense of Sugawara with finitely generated mod cohomology has the homotopy type of a finite product of , and for .

     

Bloch constants in several variables

We give lower estimates for Bloch's constant for quasiregular holomorphic mappings. A holomorphic mapping of the unit ball into is -quasiregular if it maps infinitesimal spheres to infinitesimal ellipsoids whose major axes are less than or equal to times their minor axes. We show that if is a -quasiregular holomorphic mapping with the normalization then the image contains a schlicht ball of radius at least This result is best possible in terms of powers of Also, we extend to several variables an analogous result of Landau for bounded holomorphic functions in the unit disk.

     

Serre's generalization of Nagao's theorem: An elementary approach

Let be a smooth projective curve over a field . For each closed point of let be the coordinate ring of the affine curve obtained by removing from . Serre has proved that is isomorphic to the fundamental group, , of a graph of groups , where is a tree with at most one non-terminal vertex. Moreover the subgroups of attached to the terminal vertices of are in one-one correspondence with the elements of , the ideal class group of . This extends an earlier result of Nagao for the simplest case .

Serre's proof is based on applying the theory of groups acting on trees to the quotient graph , where is the associated Bruhat-Tits building. To determine he makes extensive use of the theory of vector bundles (of rank 2) over . In this paper we determine using a more elementary approach which involves substantially less algebraic geometry.

The subgroups attached to the edges of are determined (in part) by a set of positive integers , say. In this paper we prove that is bounded, even when Cl is infinite. This leads, for example, to new free product decomposition results for certain principal congruence subgroups of , involving unipotent and elementary matrices.

     

Comparing Heegaard and JSJ structures of orientable 3-manifolds

The Heegaard genus of an irreducible closed orientable -manifold puts a limit on the number and complexity of the pieces that arise in the Jaco-Shalen-Johannson decomposition of the manifold by its canonical tori. For example, if of the complementary components are not Seifert fibered, then . This generalizes work of Kobayashi. The Heegaard genus also puts explicit bounds on the complexity of the Seifert pieces. For example, if the union of the Seifert pieces has base space and exceptional fibers, then .

     

On the telescopic homotopy theory of spaces

In telescopic homotopy theory, a space or spectrum is approximated by a tower of localizations , , taking account of -periodic homotopy groups for progressively higher . For each , we construct a telescopic Kuhn functor carrying a space to a spectrum with the same -periodic homotopy groups, and we construct a new functor left adjoint to . Using these functors, we show that the th stable monocular homotopy category (comprising the th fibers of stable telescopic towers) embeds as a retract of the th unstable monocular homotopy category in two ways: one giving infinite loop spaces and the other giving ``infinite -suspension spaces.' We deduce that Ravenel's stable telescope conjectures are equivalent to unstable telescope conjectures. In particular, we show that the failure of Ravenel's th stable telescope conjecture implies the existence of highly connected infinite loop spaces with trivial Johnson-Wilson -homology but nontrivial -periodic homotopy groups, showing a fundamental difference between the unstable chromatic and telescopic theories. As a stable chromatic application, we show that each spectrum is -equivalent to a suspension spectrum. As an unstable chromatic application, we determine the -localizations and -localizations of infinite loop spaces in terms of -localizations of spectra under suitable conditions. We also determine the -localizations and -localizations of arbitrary Postnikov -spaces.

     

The cohomology of certain Hopf algebras associated with -groups

We study the cohomology of a locally finite, connected, cocommutative Hopf algebra over . Specifically, we are interested in those algebras for which is generated as an algebra by and . We shall call such algebras semi-Koszul. Given a central extension of Hopf algebras with monogenic and semi-Koszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for to be semi-Koszul. Special attention is given to the case in which is the restricted universal enveloping algebra of the Lie algebra obtained from the mod- lower central series of a -group. We show that the algebras arising in this way from extensions by of an abelian -group are semi-Koszul. Explicit calculations are carried out for algebras arising from rank 2 -groups, and it is shown that these are all semi-Koszul for .

     

Metrics and embeddings of generalizations of Thompson's group

The distance from the origin in the word metric for generalizations of Thompson's group is quasi-isometric to the number of carets in the reduced rooted tree diagrams representing the elements of . This interpretation of the metric is used to prove that every admits a quasi-isometric embedding into every , and also to study the behavior of the shift maps under these embeddings.

     

Sharp Sobolev inequalities with lower order remainder terms

Given a smooth compact Riemannian -manifold , this paper deals with the sharp Sobolev inequality corresponding to the embedding of in where the remainder term is replaced by a lower order term.

     

On the invariant faces associated with a cone-preserving map

For an nonnegative matrix , an isomorphism is obtained between the lattice of initial subsets (of ) for and the lattice of -invariant faces of the nonnegative orthant . Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If leaves invariant a polyhedral cone , then for each distinguished eigenvalue of for , there is a chain of distinct -invariant join-irreducible faces of , each containing in its relative interior a generalized eigenvector of corresponding to (referred to as semi-distinguished -invariant faces associated with ), where is the maximal order of distinguished generalized eigenvectors of corresponding to , but there is no such chain with more than members. We introduce the important new concepts of semi-distinguished -invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.

     

On modules of finite upper rank

For a group and a prime , the upper -rank of is the supremum of the sectional -ranks of all finite quotients of . It is unknown whether, for a finitely generated group , these numbers can be finite but unbounded as ranges over all primes. The conjecture that this cannot happen if is soluble is reduced to an analogous `relative' conjecture about the upper -ranks of a `quasi-finitely-generated' module for a soluble minimax group . The main result establishes a special case of this relative conjecture, namely when the module is finitely generated and the minimax group is abelian-by-polycyclic. The proof depends on generalising results of Roseblade on group rings of polycyclic groups to group rings of soluble minimax groups. (If true in general, the above-stated conjecture would imply the truth of Lubotzky's `Gap Conjecture' for subgroup growth, in the case of soluble groups; the Gap Conjecture is known to be false for non-soluble groups.)     

Computing the homology of Koszul complexes

Let be a commutative ring and an ideal in which is locally generated by a regular sequence of length . Then, each f. g. projective -module has an -projective resolution of length . In this paper, we compute the homology of the -th Koszul complex associated with the homomorphism for all , if . This computation yields a new proof of the classical Adams-Riemann-Roch formula for regular closed immersions which does not use the deformation to the normal cone any longer. Furthermore, if , we compute the homology of the complex where and denote the functors occurring in the Dold-Kan correspondence.