On almost one-to-one maps

A continuous map of topological spaces is said to be almost -to- if the set of the points such that is dense in ; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and -compact spaces (e.g., -manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if is a minimal self-mapping of a 2-manifold , then point preimages under are tree-like continua and either is a union of 2-tori, or is a union of Klein bottles permuted by .

     

A monoidal approach to splitting morphisms of bialgebras

The main goal of this paper is to investigate the structure of Hopf algebras with the property that either its Jacobson radical is a Hopf ideal or its coradical is a subalgebra. Let us consider a Hopf algebra such that its Jacobson radical is a nilpotent Hopf ideal and is a semisimple algebra. We prove that the canonical projection of on has a section which is an -colinear algebra map. Furthermore, if is cosemisimple too, then we can choose this section to be an -bicolinear algebra morphism. This fact allows us to describe as a `generalized bosonization' of a certain algebra in the category of Yetter-Drinfeld modules over . As an application we give a categorical proof of Radford's result about Hopf algebras with projections. We also consider the dual situation. Let be a bialgebra such that its coradical is a Hopf sub-bialgebra with antipode. Then there is a retraction of the canonical injection of into which is an -linear coalgebra morphism. Furthermore, if is semisimple too, then we can choose this retraction to be an -bilinear coalgebra morphism. Then, also in this case, we can describe as a `generalized bosonization' of a certain coalgebra in the category of Yetter-Drinfeld modules over .

     

Stationary isothermic surfaces and uniformly dense domains

We establish a relationship between stationary isothermic surfaces and uniformly dense domains. A stationary isothermic surface is a level surface of temperature which does not evolve with time. A domain in the -dimensional Euclidean space is said to be uniformly dense in a surface of codimension if, for every small the volume of the intersection of with a ball of radius and center does not depend on for

We prove that the boundary of every uniformly dense domain which is bounded (or whose complement is bounded) must be a sphere. We then examine a uniformly dense domain with unbounded boundary , and we show that the principal curvatures of satisfy certain identities.

The case in which the surface coincides with is particularly interesting. In fact, we show that, if the boundary of a uniformly dense domain is connected, then (i) if , it must be either a circle or a straight line and (ii) if it must be either a sphere, a spherical cylinder or a minimal surface. We conclude with a discussion on uniformly dense domains whose boundary is a minimal surface.

     

Frobenius morphisms and representations of algebras

By introducing Frobenius morphisms on algebras and their modules over the algebraic closure of the finite field of elements, we establish a relation between the representation theory of over and that of the -fixed point algebra over . More precisely, we prove that the category    mod- of finite-dimensional -modules is equivalent to the subcategory of finite-dimensional -stable -modules, and, when is finite dimensional, we establish a bijection between the isoclasses of indecomposable -modules and the -orbits of the isoclasses of indecomposable -modules. Applying the theory to representations of quivers with automorphisms, we show that representations of a modulated quiver (or a species) over can be interpreted as -stable representations of the corresponding quiver over . We further prove that every finite-dimensional hereditary algebra over is Morita equivalent to some , where is the path algebra of a quiver over and is induced from a certain automorphism of . A close relation between the Auslander-Reiten theories for and is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of is obtained by ``folding" the Auslander-Reiten quiver of . Finally, by taking Frobenius fixed points, we are able to count the number of indecomposable representations of a modulated quiver over with a given dimension vector and to generalize Kac's theorem for all modulated quivers and their associated Kac-Moody algebras defined by symmetrizable generalized Cartan matrices.

     

Resolutions for metrizable compacta in extension theory

We prove a -resolution theorem for simply connected CW- complexes in extension theory in the class of metrizable compacta . This means that if is a connected CW-complex, is an abelian group, , , for , and (in the sense of extension theory, that is, is an absolute extensor for ), then there exists a metrizable compactum and a surjective map such that:

(a) is -acyclic,

(b) , and

(c) .

This implies the -resolution theorem for arbitrary abelian groups for cohomological dimension when . Thus, in case is an Eilenberg-MacLane complex of type , then (c) becomes .

If in addition , then (a) can be replaced by the stronger statement,

(aa) is -acyclic.

To say that a map is -acyclic means that for each , every map of the fiber to is nullhomotopic.

     

Rotation topological factors of minimal -actions on the Cantor set

In this paper we study conditions under which a free minimal -action on the Cantor set is a topological extension of the action of rotations, either on the product of -tori or on a single -torus . We extend the notion of linearly recurrent systems defined for -actions on the Cantor set to -actions, and we derive in this more general setting a necessary and sufficient condition, which involves a natural combinatorial data associated with the action, allowing the existence of a rotation topological factor of one of these two types.

     

A general theory of almost convex functions

Let be the standard -dimensional simplex and let . Then a function with domain a convex set in a real vector space is -almost convex iff for all and the inequality

holds. A detailed study of the properties of -almost convex functions is made. If contains at least one point that is not a vertex, then an extremal -almost convex function is constructed with the properties that it vanishes on the vertices of and if is any bounded -almost convex function with on the vertices of , then for all . In the special case , the barycenter of , very explicit formulas are given for and . These are of interest, as and are extremal in various geometric and analytic inequalities and theorems.

     

Bounded Hochschild cohomology of Banach algebras with a matrix-like structure

Let be a unital Banach algebra. A projection in which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal in . In this set-up we prove a theorem to the effect that the bounded cohomology vanishes for all . The hypotheses of this theorem involve (i) strong H-unitality of , (ii) a growth condition on diagonal matrices in , and (iii) an extension of in by an amenable Banach algebra. As a corollary we show that if is an infinite dimensional Banach space with the bounded approximation property, is an infinite dimensional -space, and is the Banach algebra of approximable operators on , then for all .

     

The invariant factors of the incidence matrices of points and subspaces in and

We determine the Smith normal forms of the incidence matrices of points and projective -dimensional subspaces of and of the incidence matrices of points and -dimensional affine subspaces of for all , , and arbitrary prime power .

     

Compressions of resolvents and maximal radius of regularity

Suppose that is left invertible in for all , where is an open subset of the complex plane. Then an operator-valued function is a left resolvent of in if and only if has an extension , the resolvent of which is a dilation of of a particular form. Generalized resolvents exist on every open set , with included in the regular domain of . This implies a formula for the maximal radius of regularity of in terms of the spectral radius of its generalized inverses. A solution to an open problem raised by
J. Zemánek is obtained.

     

A generalization of Marshall's equivalence relation

For prime and for a field containing a root of unity of order , we generalize Marshall's equivalence relation on orderings to arbitrary subgroups of of index . The equivalence classes then correspond to free pro- factors of the maximal pro- Galois group of . We generalize to this setting results of Jacob on the maximal pro- Galois group of a Pythagorean field.

     

On initial segment complexity and degrees of randomness

One approach to understanding the fine structure of initial segment complexity was introduced by Downey, Hirschfeldt and LaForte. They define to mean that . The equivalence classes under this relation are the -degrees. We prove that if is -random, then and have no upper bound in the -degrees (hence, no join). We also prove that -randomness is closed upward in the -degrees. Our main tool is another structure intended to measure the degree of randomness of real numbers: the -degrees. Unlike the -degrees, many basic properties of the -degrees are easy to prove. We show that implies , so some results can be transferred. The reverse implication is proved to fail. The same analysis is also done for , the analogue of for plain Kolmogorov complexity.

Two other interesting results are included. First, we prove that for any , a -random real computable from a --random real is automatically --random. Second, we give a plain Kolmogorov complexity characterization of -randomness. This characterization is related to our proof that implies .

     

The braid index is not additive for the connected sum of 2-knots

Any -dimensional knot can be presented in a braid form, and its braid index, , is defined. For the connected sum of -knots and , it is easily seen that holds. Birman and Menasco proved that the braid index (minus one) is additive for the connected sum of -dimensional knots; the equality holds for -knots. We prove that the equality does not hold for -knots unless or is a trivial -knot. We also prove that the -knot obtained from a granny knot by Artin's spinning is of braid index , and there are infinitely many -knots of braid index .

     

Prime geodesic theorem for higher-dimensional hyperbolic manifold

For a -dimensional hyperbolic manifold , we consider an estimate of the error term of the prime geodesic theorem. Put the fundamental group of to be a discrete subgroup of with cofinite volume. When the contribution of the discrete spectrum of the Laplace-Beltrami operator is larger than that of the continuous spectrum in Weyl's law, we obtained a lower estimate as goes to .

     

Vertex operators for twisted quantum affine algebras

We construct explicitly the -vertex operators (intertwining operators) for the level one modules of the classical quantum affine algebras of twisted types using interacting bosons, where for (), for , for (), and for (). A perfect crystal graph for is constructed as a by-product.

     

Filtrations in semisimple Lie algebras, I

In this paper, we study the maximal bounded -filtrations of a complex semisimple Lie algebra . Specifically, we show that if is simple of classical type , , or , then these filtrations correspond uniquely to a precise set of linear functionals on its root space. We obtain partial, but not definitive, results in this direction for the remaining exceptional algebras. Maximal bounded filtrations were first introduced in the context of classifying the maximal graded subalgebras of affine Kac-Moody algebras, and the maximal graded subalgebras of loop toroidal Lie algebras. Indeed, our main results complete this classification in most cases. Finally, we briefly discuss the analogous question for bounded filtrations with respect to other Archimedean ordered groups.

     

Algebraic Goodwillie calculus and a cotriple model for the remainder

Goodwillie has defined a tower of approximations for a functor from spaces to spaces that is analogous to the Taylor series of a function. His order approximation at a space depends on the values of on coproducts of large suspensions of the space: .

We define an ``algebraic' version of the Goodwillie tower, , that depends only on the behavior of on coproducts of . When is a functor to connected spaces or grouplike -spaces, the functor is the base of a fibration

whose fiber is the simplicial space associated to a cotriple built from the cross effect of the functor . In a range in which commutes with realizations (for instance, when is the identity functor of spaces), the algebraic Goodwillie tower agrees with the ordinary (topological) Goodwillie tower, so this theory gives a way of studying the Goodwillie approximation to a functor in many interesting cases.

     

Poisson structures on complex flag manifolds associated with real forms

For a complex semisimple Lie group and a real form we define a Poisson structure on the variety of Borel subgroups of with the property that all -orbits in as well as all Bruhat cells (for a suitable choice of a Borel subgroup of ) are Poisson submanifolds. In particular, we show that every non-empty intersection of a -orbit and a Bruhat cell is a regular Poisson manifold, and we compute the dimension of its symplectic leaves.

     

Length, multiplicity, and multiplier ideals

Let be an -dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an -primary ideal of , the relationship between the singularities of the scheme defined by and those defined by the multiplier ideals , with varying in , are quantified in this paper by showing that the Samuel multiplicity of satisfies whenever . This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustata and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.

     

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.