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.

     

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.

     

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.

     

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 .

     

A universal continuum of weight

We prove that every continuum of weight  is a continuous image of the Cech-Stone-remainder  of the real line. It follows that under  the remainder of the half line is universal among the continua of weight  -- universal in the `mapping onto' sense.

We complement this result by showing that 1) under  every continuum of weight less than  is a continuous image of , 2) in the Cohen model the long segment of length  is not a continuous image of , and 3)  implies that is not a continuous image of , whenever  is a -saturated ultrafilter.

We also show that a universal continuum can be gotten from a -saturated ultrafilter on , and that it is consistent that there is no universal continuum of weight  .

     

On supports and associated primes of modules over the enveloping algebras of nilpotent Lie algebras

Let be a nilpotent Lie algebra, over a field of characteristic zero, and its universal enveloping algebra. In this paper we study: (1) the prime ideal structure of related to finitely generated -modules , and in particular the set of associated primes for such (note that now is equal to the set of annihilator primes for ); (2) the problem of nontriviality for the modules when is a (maximal) prime of , and in particular when is the augmentation ideal of . We define the support of , as a natural generalization of the same notion from commutative theory, and show that it is the object of primary interest when dealing with (2). We also introduce and study the reduced localization and the reduced support, which enables to better understand the set . We prove the following generalization of a stability result given by W. Casselman and M. S. Osborne in the case when , as in the theorem, are abelian. We also present some of its interesting consequences.

Theorem. Let be a finite-dimensional Lie algebra over a field of characteristic zero, and an ideal of ; denote by the universal enveloping algebra of . Let be a -module which is finitely generated as an -module. Then every annihilator prime of , when is regarded as a -module, is -stable for the adjoint action of on .

     

On the number of -equivalent non-isomorphic models

We prove that if is consistent then is consistent with the following statement: There is for every a model of cardinality which is -equivalent to exactly non-isomorphic models of cardinality . In order to get this result we introduce ladder systems and colourings different from the ``standard' counterparts, and prove the following purely combinatorial result: For each prime number and positive integer it is consistent with that there is a ``good' ladder system having exactly pairwise nonequivalent colourings.

     

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

     

Groups with two extreme character degrees and their normal subgroups

We study the finite groups for which the set of irreducible complex character degrees consists of the two most extreme possible values, that is, and . We are easily reduced to finite -groups, for which we derive the following group theoretical characterization: they are the -groups such that is a square and whose only normal subgroups are those containing or contained in . By analogy, we also deal with -groups such that is not a square, and we prove that if and only if a similar property holds: for any , either or . The proof of these results requires a detailed analysis of the structure of the -groups with any of the conditions above on normal subgroups, which is interesting for its own sake. It is especially remarkable that these groups have small nilpotency class and that, if the nilpotency class is greater than , then the index of the centre is small, and in some cases we may even bound the order of .

     

A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of

For a domain contained in a hemisphere of the -dimensional sphere we prove the optimal result for the ratio of its first two Dirichlet eigenvalues where , the symmetric rearrangement of in , is a geodesic ball in having the same -volume as . We also show that for geodesic balls of geodesic radius less than or equal to is an increasing function of which runs between the value for (this is the Euclidean value) and for . Here denotes the th positive zero of the Bessel function . This result generalizes the Payne-Pólya-Weinberger conjecture, which applies to bounded domains in Euclidean space and which we had proved earlier. Our method makes use of symmetric rearrangement of functions and various technical properties of special functions. We also prove that among all domains contained in a hemisphere of and having a fixed value of the one with the maximal value of is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for . Various other results for and of geodesic balls in are proved in the course of our work.

     

Local derivations on -algebras are derivations

Kadison has shown that local derivations from a von Neumann algebra into any dual bimodule are derivations. In this paper we extend this result to local derivations from any -algebra into any Banach -bimodule . Most of the work is involved with establishing this result when is a commutative -algebra with one self-adjoint generator. A known result of the author about Jordan derivations then completes the argument. We show that these results do not extend to the algebra of continuously differentiable functions on . We also give an automatic continuity result, that is, we show that local derivations on -algebras are continuous even if not assumed a priori to be so.

     

-actions whose fixed data has a section

Given a collection of real vector bundles over a closed manifold , suppose that, for some is of the form , where is the trivial one-dimensional bundle. In this paper we prove that if is the fixed data of a -action, then the same is true for the Whitney sum obtained from by replacing by . This stability property is well-known for involutions. Together with techniques previously developed, this result is used to describe, up to bordism, all possible -actions fixing the disjoint union of an even projective space and a point.     

The Lipschitz continuity of the distance function to the cut locus

Let be a closed submanifold of a complete smooth Riemannian manifold and the total space of the unit normal bundle of . For each , let denote the distance from to the cut point of on the geodesic with the velocity vector The continuity of the function on is well known. In this paper we prove that is locally Lipschitz on which is bounded; in particular, if and are compact, then is globally Lipschitz on . Therefore, the canonical interior metric may be introduced on each connected component of the cut locus of and this metric space becomes a locally compact and complete length space.

     

Properties of Anick's spaces

We prove three useful properties of Anick's space  . First, at odd primes a map from into a homotopy commutative, homotopy associative -space can be extended to a unique -map from into . Second, at primes larger than , is itself homotopy commutative and homotopy associative. And third, the first two properties combine to show that the order of the identity map on  is .

     

Simple holonomic modules over rings of differential operators with regular coefficients of Krull dimension 2

Let be an algebraically closed field of characteristic zero. Let be the ring of (-linear) differential operators with coefficients from a regular commutative affine domain of Krull dimension which is the tensor product of two regular commutative affine domains of Krull dimension . Simple holonomic -modules are described. Let a -algebra be a regular affine commutative domain of Krull dimension and be the ring of differential operators with coefficients from . We classify (up to irreducible elements of a certain Euclidean domain) simple -modules (the field is not necessarily algebraically closed).

     

Relative Embedding Problems

We consider Galois embedding problems such that a Galois embedding problem is solvable, where is a Galois subextension of . For such embedding problems with abelian kernel, we prove a reduction theorem, first in the general case of commutative -algebras, then in the more specialized field case. We demonstrate with examples of dihedral embedding problems that the reduced embedding problem is frequently of smaller order. We then apply these results to the theory of obstructions to central embedding problems, considering a notion of quotients of central embedding problems, and classify the infinite towers of metacyclic -groups to which the reduction theorem applies.

     

On the representation of unity by binary cubic forms

If is a binary cubic form with integer coefficients such that has at least two distinct complex roots, then the equation possesses at most ten solutions in integers and , nine if has a nontrivial automorphism group. If, further, is reducible over , then this equation has at most solutions, unless is equivalent under -action to either or . The proofs of these results rely upon the method of Thue-Siegel as refined by Evertse, together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational Diophantine approximation. Along the way, we completely solve all Thue equations for cubic and irreducible of positive discriminant . As corollaries, we obtain bounds for the number of solutions to more general cubic Thue equations of the form and to Mordell's equation , where and are nonzero integers.

     

Infinite convolution products and refinable distributions on Lie groups

Sufficient conditions for the convergence in distribution of an infinite convolution product of measures on a connected Lie group with respect to left invariant Haar measure are derived. These conditions are used to construct distributions that satisfy where is a refinement operator constructed from a measure and a dilation automorphism . The existence of implies is nilpotent and simply connected and the exponential map is an analytic homeomorphism. Furthermore, there exists a unique minimal compact subset such that for any open set containing and for any distribution on with compact support, there exists an integer such that implies If is supported on an -invariant uniform subgroup then is related, by an intertwining operator, to a transition operator on Necessary and sufficient conditions for to converge to , and for the -translates of to be orthogonal or to form a Riesz basis, are characterized in terms of the spectrum of the restriction of to functions supported on

     

On syzygies of abelian varieties

In this paper we prove the following result: Let be a complex torus and a normally generated line bundle on ; then, for every , the line bundle satisfies Property of Green-Lazarsfeld.

     

Transfers of Chern classes in BP-cohomology and Chow rings

The -module structure of for extraspecial -groups is studied using transfer and Chern classes. These give rise to -torsion elements in the kernel of the cycle map from the Chow ring to ordinary cohomology first obtained by Totaro.