Trees and valuation rings

A subring of a division algebra is called a valuation ring of if or holds for all nonzero in . The set of all valuation rings of is a partially ordered set with respect to inclusion, having as its maximal element. As a graph is a rooted tree (called the valuation tree of ), and in contrast to the commutative case, may have finitely many but more than one vertices. This paper is mainly concerned with the question of whether each finite, rooted tree can be realized as a valuation tree of a division algebra , and one main result here is a positive answer to this question where can be chosen as a quaternion division algebra over a commutative field.

     

Homology manifold bordism

The Bryant-Ferry-Mio-Weinberger surgery exact sequence for compact homology manifolds of dimension is used to obtain transversality, splitting and bordism results for homology manifolds, generalizing previous work of Johnston.

First, we establish homology manifold transversality for submanifolds of dimension : if is a map from an -dimensional homology manifold to a space , and is a subspace with a topological -block bundle neighborhood, and , then is homology manifold -cobordant to a map which is transverse to , with an -dimensional homology submanifold.

Second, we obtain a codimension splitting obstruction in the Wall -group for a simple homotopy equivalence from an -dimensional homology manifold to an -dimensional Poincaré space with a codimension Poincaré subspace with a topological normal bundle, such that if (and for only if) splits at up to homology manifold -cobordism.

Third, we obtain the multiplicative structure of the homology manifold bordism groups .

     

Sums of squares of regular functions on real algebraic varieties

Let be an affine algebraic variety over (or any other real closed field ). We ask when it is true that every positive semidefinite (psd) polynomial function on is a sum of squares (sos). We show that for the answer is always negative if has a real point. Also, if is a smooth non-rational curve all of whose points at infinity are real, the answer is again negative. The same holds if is a smooth surface with only real divisors at infinity. The ``compact' case is harder. We completely settle the case of smooth curves of genus : If such a curve has a complex point at infinity, then every psd function is sos, provided the field is archimedean. If is not archimedean, there are counter-examples of genus .

     

A bracket power characterization of analytic spread one ideals

The main theorem characterizes, in terms of bracket powers, analytic spread one ideals in local rings. Specifically, let be regular nonunits in a local (Noetherian) ring and assume that , the integral closure of , where . Then the main result shows that for all but finitely many units in that are non-congruent modulo and for all large integers and it holds that for and not divisible by , where is the -th bracket power of . And, conversely, if there exist positive integers , , and such that has a basis such that , then has analytic spread one.

     

Specializations of Brauer classes over algebraic function fields

Let be either a number field or a field finitely generated of transcendence degree over a Hilbertian field of characteristic 0, let be the rational function field in one variable over , and let . It is known that there exist infinitely many such that the specialization induces a specialization , where has exponent equal to that of . Now let be a finite extension of and let . We give sufficient conditions on and for there to exist infinitely many such that the specialization has an extension to inducing a specialization , the residue field of , where has exponent equal to that of . We also give examples to show that, in general, such need not exist.

     

Unipotent groups associated to reduced curves

Let be a curve defined over an algebraically closed field with 0$">. Assume that is reduced. In this paper we study the unipotent part of the Jacobian . In particular, we prove that if is large in terms of the dimension of , then is isomorphic to a product of additive groups .

     

Invariant foliations near normally hyperbolic invariant manifolds for semiflows

Let be a compact manifold which is invariant and normally hyperbolic with respect to a semiflow in a Banach space. Then in an -neighborhood of there exist local center-stable and center-unstable manifolds and , respectively. Here we show that and may each be decomposed into the disjoint union of submanifolds (leaves) in such a way that the semiflow takes leaves into leaves of the same collection. Furthermore, each leaf intersects in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time.

     

Homology decompositions for classifying spaces of compact Lie groups

Let be a prime number and be a compact Lie group. A homology decomposition for the classifying space is a way of building up to mod homology as a homotopy colimit of classifying spaces of subgroups of . In this paper we develop techniques for constructing such homology decompositions. Jackowski, McClure and Oliver (Homotopy classification of self-maps of BG via -actions, Ann. of Math. 135 (1992), 183-270) construct a homology decomposition of by classifying spaces of -stubborn subgroups of . Their decomposition is based on the existence of a finite-dimensional mod acyclic --complex with restricted set of orbit types. We apply our techniques to give a parallel proof of the -stubborn decomposition of which does not use this geometric construction.

     

Principal curvatures of isoparametric hypersurfaces in

Let be an isoparametric hypersurface in , and the inverse image of under the Hopf map. By using the relationship between the eigenvalues of the shape operators of and , we prove that is homogeneous if and only if either or is constant, where is the number of distinct principal curvatures of and is the number of non-horizontal eigenspaces of the shape operator on .

     

Stability theory, permutations of indiscernibles, and embedded finite models

We show that the expressive power of first-order logic over finite models embedded in a model is determined by stability-theoretic properties of . In particular, we show that if is stable, then every class of finite structures that can be defined by embedding the structures in , can be defined in pure first-order logic. We also show that if does not have the independence property, then any class of finite structures that can be defined by embedding the structures in , can be defined in first-order logic over a dense linear order. This extends known results on the definability of classes of finite structures and ordered finite structures in the setting of embedded finite models. These results depend on several results in infinite model theory. Let be a set of indiscernibles in a model and suppose is elementarily equivalent to where is -saturated. If is stable and is saturated, then every permutation of extends to an automorphism of and the theory of is stable. Let be a sequence of -indiscernibles in a model , which does not have the independence property, and suppose is elementarily equivalent to where is a complete dense linear order and is -saturated. Then -types over are order-definable and if is -saturated, every order preserving permutation of can be extended to a back-and-forth system.

     

Local structure of Schelter-Procesi smooth orders

In this paper we give the étale local classification of Schelter-Procesi smooth orders in central simple algebras. In particular, we prove that if is a central simple -algebra of dimension , where is a field of trancendence degree , then there are only finitely many étale local classes of smooth orders in . This result is a non-commutative generalization of the fact that a smooth variety is analytically a manifold, and so has only one type of étale local behaviour.

     

On the endomorphism monoids of (uniquely) complemented lattices

Let be a lattice with and . An endomorphism of is a -endomorphism, if it satisfies and . The -endomorphisms of form a monoid. In 1970, the authors proved that every monoid can be represented as the -endomorphism monoid of a suitable lattice with and . In this paper, we prove the stronger result that the lattice with a given -endomorphism monoid can be constructed as a uniquely complemented lattice; moreover, if is finite, then can be chosen as a finite complemented lattice.

     

The Farrell-Jones Isomorphism Conjecture for finite covolume hyperbolic actions and the algebraic -theory of Bianchi groups

We prove the Farrell-Jones Isomorphism Conjecture for groups acting properly discontinuously via isometries on (real) hyperbolic -space with finite volume orbit space. We then apply this result to show that, for any Bianchi group , , , and vanish for .

     

Steiner type formulae and weighted measures of singularities for semi-convex functions

For a given convex (semi-convex) function , defined on a nonempty open convex set , we establish a local Steiner type formula, the coefficients of which are nonnegative (signed) Borel measures. We also determine explicit integral representations for these coefficient measures, which are similar to the integral representations for the curvature measures of convex bodies (and, more generally, of sets with positive reach). We prove that, for , the -th coefficient measure of the local Steiner formula for , restricted to the set of -singular points of , is absolutely continuous with respect to the -dimensional Hausdorff measure, and that its density is the -dimensional Hausdorff measure of the subgradient of .

As an application, under the assumptions that is convex and Lipschitz, and is bounded, we get sharp estimates for certain weighted Hausdorff measures of the sets of -singular points of . Such estimates depend on the Lipschitz constant of and on the quermassintegrals of the topological closure of .

     

Embeddings in generalized manifolds

We prove that a ()-connected map from a compact PL -manifold to a generalized -manifold with the disjoint disks property, , is homotopic to a tame embedding. There is also a controlled version of this result, as well as a version for noncompact and proper maps that are properly ()-connected. The techniques developed lead to a general position result for arbitrary maps , , and a Whitney trick for separating submanifolds of that have intersection number 0, analogous to the well-known results when is a manifold.

     

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.

     

An equivariant Brauer semigroup and the symmetric imprimitivity theorem

Suppose that is a second countable locally compact transformation group. We let denote the set of Morita equivalence classes of separable dynamical systems where is a -algebra and is compatible with the given -action on . We prove that is a commutative semigroup with identity with respect to the binary operation for an appropriately defined balanced tensor product on -algebras. If and act freely and properly on the left and right of a space , then we prove that and are isomorphic as semigroups. If the isomorphism maps the class of to the class of , then is Morita equivalent to .

     

On the dimension of the product of two compacta and the dimension of their intersection in general position in Euclidean space

For every two compact metric spaces and , both with dimension at most , there are dense -subsets of mappings and with .

     

Definably simple groups in o-minimal structures

Let be a group definable in an o-minimal structure . A subset of is -definable if is definable in the structure (while definable means definable in the structure ). Assume has no -definable proper subgroup of finite index. In this paper we prove that if has no nontrivial abelian normal subgroup, then is the direct product of -definable subgroups such that each is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin's conjecture.