An isometry theorem for quadratic differentials on Riemann surfaces of finite genus

Assume both and are Riemann surfaces which are subsets of compact Riemann surfaces and respectively, and that the set has infinitely many points. We show that the only surjective complex linear isometries between the spaces of integrable holomorphic quadratic differentials on and are the ones induced by conformal homeomorphisms and complex constants of modulus 1. It follows that every biholomorphic map from the Teichmüller space of onto the Teichmüller space of is induced by some quasiconformal map of onto . Consequently we can find an uncountable set of Riemann surfaces whose Teichmüller spaces are not biholomorphically equivalent.

     

Cohomology of the complement of a free divisor

We prove that if is a ``strongly quasihomogeneous" free divisor in the Stein manifold , and is its complement, then the de Rham cohomology of can be computed as the cohomology of the complex of meromorphic differential forms on with logarithmic poles along , with exterior derivative. The class of strongly quasihomogeneous free divisors, introduced here, includes free hyperplane arrangements and the discriminants of stable mappings in Mather's nice dimensions (and in particular the discriminants of Coxeter groups).

     

Geometric Isomorphisms between Infinite Dimensional Teichmüller Spaces

Let and be the interiors of bordered Riemann surfaces with finitely generated fundamental groups and nonempty borders. We prove that every holomorphic isomorphism of the Teichmüller space of onto the Teichmüller space of is induced by a quasiconformal homeomorphism of onto . These Teichmüller spaces are not finite dimensional and their groups of holomorphic automorphisms do not act properly discontinuously, so the proof presents difficulties not present in the classical case. To overcome them we study weak continuity properties of isometries of the tangent spaces to Teichmüller space and special properties of Teichmüller disks.

     

Plane Cremona maps, exceptional curves and roots

We address three different questions concerning exceptional and root divisors (of arithmetic genus zero and of self-intersection and , respectively) on a smooth complex projective surface which admits a birational morphism to . The first one is to find criteria for the properness of these divisors, that is, to characterize when the class of is in the -orbit of the class of the total transform of some point blown up by if is exceptional, or in the -orbit of a simple root if is root, where is the Weyl group acting on ; we give an arithmetical criterion, which adapts an analogous criterion suggested by Hudson for homaloidal divisors, and a geometrical one. Secondly, we prove that the irreducibility of the exceptional or root divisor is a necessary and sufficient condition in order that could be transformed into a line by some plane Cremona map, and in most cases for its contractibility. Finally, we provide irreducibility criteria for proper homaloidal, exceptional and effective root divisors.

     

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.

     

The image of the Thom map for Eilenberg-MacLane spaces

Fundamental classes in cohomology of Eilenberg-MacLane spaces are defined. The image of the Thom map from cohomology to mod- cohomology is determined for arbitrary Eilenberg-MacLane spaces. This image is a polynomial subalgebra generated by infinitely many elements obtained by applying a maximum number of Milnor primitives to the fundamental class in mod- cohomology. This subalgebra in mod cohomology is invariant under the action of the Steenrod algebra, and it is annihilated by all Milnor primitives. We also show that cohomology determines Morava cohomology for Eilenberg-MacLane spaces.

     

Absolute Borel sets and function spaces

An internal characterization of metric spaces which are absolute Borel sets of multiplicative classes is given. This characterization uses complete sequences of covers, a notion introduced by Frolík for characterizing Cech-complete spaces. We also show that the absolute Borel class of is determined by the uniform structure of the space of continuous functions ; however the case of absolute metric spaces is still open. More precisely, we prove that, for metrizable spaces and , if is a uniformly continuous surjection and is an absolute Borel set of multiplicative (resp., additive) class , , then is also an absolute Borel set of the same class. This result is new even if is a linear homeomorphism, and extends a result of Baars, de Groot, and Pelant which shows that the \v{C}ech-completeness of a metric space is determined by the linear structure of .

     

Gross spaces

A Gross space is a vector space of infinite dimension over some field , which is endowed with a symmetric bilinear form and has the property that every infinite dimensional subspace satisfies dim dim. Gross spaces over uncountable fields exist (in certain dimensions). The existence of a Gross space over countable or finite fields (in a fixed dimension not above the continuum) is independent of the axioms of ZFC. Here we continue the investigation of Gross spaces. Among other things, we show that if the cardinal invariant b equals , a Gross space in dimension exists over every infinite field, and that it is consistent that Gross spaces exist over every infinite field but not over any finite field. We also generalize the notion of a Gross space and construct generalized Gross spaces in ZFC.

     

homology over traced *-algebras

Given a unital complex *-algebra , a tracial positive linear functional on that factors through a *-representation of on Hilbert space, and an -module possessing a resolution by finitely generated projective -modules, we construct homology spaces for . Each is a Hilbert space equipped with a *-representation of , independent (up to unitary equivalence) of the given resolution of . A short exact sequence of -modules gives rise to a long weakly exact sequence of homology spaces. There is a Künneth formula for tensor products. The von Neumann dimension which is defined for -invariant subspaces of gives well-behaved Betti numbers and an Euler characteristic for with respect to and .

     

Some New Homogeneous Einstein Metrics on Symmetric Spaces

We classify homogeneous Einstein metrics on compact irreducible symmetric spaces. In particular, we consider symmetric spaces with rank, not isometric to a compact Lie group. Whenever there exists a closed proper subgroup of Isom acting transitively on we find all -homogeneous (non-symmetric) Einstein metrics on .

     

Abstract functions with continuous differences and Namioka spaces

Let be a semigroup and a topological space. Let be an Abelian topological group. The right differences of a function are defined by for . Let be continuous at the identity of for all in a neighbourhood of . We give conditions on or range under which is continuous for any topological space . We also seek conditions on under which we conclude that is continuous at for arbitrary . This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.

     

Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces

Necessary and sufficient conditions are given for the fractional integral operator to be bounded from weighted strong and weak spaces within the range into suitable weighted and Lipschitz spaces. We also characterize the weights for which can be extended to a bounded operator from weighted into a weighted Lipschitz space of order . Finally, under an additional assumption on the weight, we obtain necessary and sufficient conditions for the boundedness of between weighted Lipschitz spaces.

     

Duality and Polynomial Testing of Tree Homomorphisms

Let be a fixed digraph. We consider the -colouring problem, i.e., the problem of deciding which digraphs admit a homomorphism to . We are interested in a characterization in terms of the absence in of certain tree-like obstructions. Specifically, we say that has tree duality if, for all digraphs , is not homomorphic to if and only if there is an oriented tree which is homomorphic to but not to . We prove that if has tree duality then the -colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the -property studied by Gutjahr, Welzl, and Woeginger.

We then focus on the case when itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads for which the -colouring problem is -complete. We contrast these with several families of oriented triads which have tree duality, or bounded treewidth duality, and hence polynomial -colouring problems. If , then no oriented triad with an -complete -colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad . We prove that none of the oriented triads with -complete -colouring problems given in the companion paper has tree duality.

     

Recognizing constant curvature discrete groups in dimension 3

We characterize those discrete groups which can act properly discontinuously, isometrically, and cocompactly on hyperbolic -space in terms of the combinatorics of the action of on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively curved (in the large) (that is, Gromov hyperbolic), the combinatorial Riemann mapping theorem, and the Sullivan-Tukia theorem on groups which act uniformly quasiconformally on the -sphere.

     

Widths of Subgroups

We say that the width of an infinite subgroup in is if there exists a collection of essentially distinct conjugates of such that the intersection of any two elements of the collection is infinite and is maximal possible. We define the width of a finite subgroup to be . We prove that a quasiconvex subgroup of a negatively curved group has finite width. It follows that geometrically finite surfaces in closed hyperbolic -manifolds satisfy the -plane property for some .

     

Approximation of the equilibrium distribution by distributions of equal point charges with minimal energy

Let denote the classical equilibrium distribution (of total charge ) on a convex or -smooth conductor in with nonempty interior. Also, let be any th order ``Fekete equilibrium distribution' on , defined by point charges at th order ``Fekete points'. (By definition such a distribution minimizes the energy for -tuples of point charges on .) We measure the approximation to by for by estimating the differences in potentials and fields,

both inside and outside the conductor . For dimension we obtain uniform estimates at distance from the outer boundary of . Observe that throughout the interior of (Faraday cage phenomenon of electrostatics), hence on the compact subsets of . For the exterior of the precise results are obtained by comparison of potentials and energies. Admissible sets have to be regular relative to capacity and their boundaries must allow good Harnack inequalities. For the passage to interior estimates we develop additional machinery, including integral representations for potentials of measures on Lipschitz boundaries and bounds on normal derivatives of interior and exterior Green functions. Earlier, one of us had considered approximations to the equilibrium distribution by arbitrary distributions of equal point charges on . In that context there is an important open problem for the sphere which is discussed at the end of the paper.

     

Extendability of Large-Scale Lipschitz Maps

Let be metric spaces, a subset of , and a large-scale lipschitz map. It is shown that possesses a large-scale lipschitz extension (with possibly larger constants) if is a Gromov hyperbolic geodesic space or the cartesian product of finitely many such spaces. No extension exists, in general, if is an infinite-dimensional Hilbert space. A necessary and sufficient condition for the extendability of a lipschitz map is given in the case when is separable and is a proper, convex geodesic space.

     

Divisors on Generic Complete Intersections in Projective Space

Let be a generic complete intersection of hypersurfaces of degree in -dimensional projective space. We study the question when a divisor on is nonrational or of general type, and give an alternative proof of a result of Ein. We also give some improvement of Ein's result in the case .

     

On the ordering of -modal cycles

The forcing relation on -modal cycles is studied. If is an -modal cycle then the -modal cycles with block structure that force form a -horseshoe above . If -modal forces , and does not have a block structure over , then forces a -horseshoe of simple extensions of .

     

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 .