-regular Witt rings

We improve Kula's bounds on the size of possible -regular Witt rings.

     

-quotients of quaternion-Kähler manifolds

The notion of symplectic reduction has been generalized to manifolds endowed with other structures, in particular to quaternion-Kähler manifolds, namely Riemannian manifolds with holonomy in . In this work we prove that the only complete quaternion-Kähler manifold with positive scalar curvature obtainable as a quaternion-Kähler quotient by a circle action is the complex Grassmannian .

     

The -theory of toric manifolds

Toric manifolds, a topological generalization of smooth projective toric varieties, are determined by an -dimensional simple convex polytope and a function from the set of codimension-one faces into the primitive vectors of an integer lattice. Their cohomology was determined by Davis and Januszkiewicz in 1991 and corresponds with the theorem of Danilov-Jurkiewicz in the toric variety case. Recently it has been shown by Buchstaber and Ray that they generate the complex cobordism ring. We use the Adams spectral sequence to compute the -theory of all toric manifolds and certain singular toric varieties.

     

Strongly -regular rings have stable range one

A ring is said to be strongly -regular if for every there exist a positive integer and such that . For example, all algebraic algebras over a field are strongly -regular. We prove that every strongly -regular ring has stable range one. The stable range one condition is especially interesting because of Evans' Theorem, which states that a module cancels from direct sums whenever has stable range one. As a consequence of our main result and Evans' Theorem, modules satisfying Fitting's Lemma cancel from direct sums.

     

The trace of jet space

The classical Whitney extension theorem describes the trace of the space of -jets generated by functions from to an arbitrary closed subset . It establishes existence of a bounded linear extension operator as well. In this paper we investigate a similar problem for the space of functions whose higher derivatives satisfy the Zygmund condition with majorant . The main result states that the vector function belongs to the corresponding trace space if the trace to every subset of cardinality , where , can be extended to a function and . The number generally speaking cannot be reduced. The Whitney theorem can be reformulated in this way as well, but with a two-pointed subset . The approach is based on the theory of local polynomial approximations and a result on Lipschitz selections of multivalued mappings.

     

Almost linearity of -bi-Lipschitz maps between real Banach spaces

Let and be real Banach spaces. A map between and is called an -bi-Lipschitz map if for all . In this note we show that if is an -bi-Lipschitz map with from onto , then is almost linear. We also show that if is a surjective -bi-Lipschitz map with , then there exists a linear isomorphism such that

where as and .

     

Compact flat manifolds with holonomy group

In this paper we construct a family of compact flat manifolds, for all dimensions , with holonomy group isomorphic to and first Betti number zero.

     

-bundles over aspherical surfaces and 4-dimensional geometries

Melvin has shown that closed 4-manifolds that arise as -bundles over closed, connected aspherical surfaces are classified up to diffeomorphism by the Stiefel-Whitney classes of the associated bundles. We show that each such 4-manifold admits one of the geometries or [depending on whether or ]. Conversely a geometric closed, connected 4-manifold of type or is the total space of an -bundle over a closed, connected aspherical surface precisely when its fundamental group is torsion free. Furthermore the total spaces of -bundles over closed, connected aspherical surfaces are all geometric. Conversely a geometric closed, connected 4-manifold is the total space of an -bundle if and only if where is torsion free.

     

Invertibility preserving linear maps on

For Banach spaces and , we show that every unital bijective invertibility preserving linear map between and is a Jordan isomorphism. The same conclusion holds for maps between and .

     

A characterization of

This work is devoted to the relationship between topological properties of a space and those of (= the space of continuous real-valued functions on , with the topology of pointwise convergence). The emphasis is on -compactness of and on location of in . In particular, -compact cosmic spaces are characterized in this way.

     

All maps of type are boundary maps

Let be a continuous map of an interval into itself having periodic points of period for all and no other periods. It is shown that every neighborhood of contains a map such that the set of periods of the periodic points of is finite. This answers a question posed by L. S. Block and W. A. Coppel.

     

On The Homotopy Type of

We consider the homotopy type of classifying spaces , where is a finite -group, and we study the question whether or not the mod cohomology of , as an algebra over the Steenrod algebra together with the associated Bockstein spectral sequence, determine the homotopy type of . This article is devoted to producing some families of finite 2-groups where cohomological information determines the homotopy type of .

     

The length of -pseudodifferential operators

We compute the length of the -algebra generated by the algebra of b-pseudodifferential operators of order on compact manifolds with corners.

     

Affine and homeomorphic embeddings into

It is shown that

(1)

a locally compact convex subset of a topological vector space that admits a sequence of continuous affine functionals separating points of affinely embeds into a Hilbert space;

(2)

an infinite-dimensional locally compact convex subset of a metric linear space has a central point;

(3)

every -compact locally convex metric linear space topologically embeds onto a pre-Hilbert space.

     

Quasisymmetric distortion and rigidity of expanding endomorphisms of

In this paper we examine a result of D. Sullivan according to which two expanding endomorphisms of the circle are conjugate as soon as they are symmetrically conjugate. We develop general a priori estimates on the local distortion of quasisymmetric mappings and combine them with the classical naive distortion lemma to present a complete proof of Sullivan's result. A new proof is offered at the end that renders unnecessary the use of Markov partitions or the control of eigenvalues at periodic points.

     

Operator ideal norms on

Let be a real number such that and its conjugate exponent . We prove that for an operator defined on with values in a Banach space, the image of the unit ball determines whether belongs to any operator ideal and its operator ideal norm. We also show that this result fails to be true in the remaining cases of . Finally we prove that when the result holds in finite dimension, the map which associates to the image of the unit ball the operator ideal norm is continuous with respect to the Hausdorff metric.

     

A fixed-point theorem for usco maps

The familiar fixed-point theorem of Kakutani is strengthened by weakening the hypotheses on the set-valued mapping. Applications are made for and decompositions of compact metric spaces.

     

The problem on domains with piecewise smooth boundaries with applications

Let be a bounded domain in such that has piecewise smooth boudnary. We discuss the solvability of the Cauchy-Riemann equation

where is a smooth -closed form with coefficients up to the bundary of , and . In particular, Equation (0.1) is solvable with smooth up to the boundary (for appropriate degree if satisfies one of the following conditions:

i)

is the transversal intersection of bounded smooth pseudoconvex domains.

ii)

where is the union of bounded smooth pseudoconvex domains and is a pseudoconvex convex domain with a piecewise smooth boundary.

iii)

where is the intersection of bounded smooth pseudoconvex domains and is a pseudoconvex domain with a piecewise smooth boundary.

The solvability of Equation (0.1) with solutions smooth up to the boundary can be used to obtain the local solvability for on domains with piecewise smooth boundaries in a pseudoconvex manifold.