Finite polynomial orbits in finitely generated domains

LetR be a commutative domain of zero characteristic and letf(X) be a polynomial with coefficients inR. It is shown that all finite orbits inR. under the mapping induced byR have their cardinalities bounded by a constant depending onR but not onf. In the case whenR is the ring of all integers in an algebraic number field this constant is effectively determined.     

On Fredholm index, transversal approximations and Quillen's geometric complex cobordism of Hilbert manifolds with some applications to flag varieties of loop groups

In [Contemp. Math. 258 (2000) 1-19], by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In this work, by using Quinn's Transversality Theorem [Proc. Sympos. Pure. Math. 15 (1970) 213-222], it will be shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It will be shown that the Thom isomorphism in this theory will be satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps will be proved. After we discuss the relation between this theory and classical cobordism, we describe some applications to the complex cobordism of flag varieties of loop groups and we do some calculations.     

Closed 1-forms with at most one zero

We prove that in any nonzero cohomology class ξ∈H¹(M;R) there always exists a closed 1-form having at most one zero.     

Riemannian geometry of finite rank positive operators

A riemannian metric is introduced in the infinite dimensional manifold Σn of positive operators with rank n<∞ on a Hilbert space H. The geometry of this manifold is studied and related to the geometry of the submanifolds Σp of positive operators with range equal to the range of a projection p (rank of p=n), and Pp of selfadjoint projections in the connected component of p. It is shown that these spaces are complete in the geodesic distance.     

Residue formulation of the Chern character on smooth manifolds

The Chern character of a complex vector bundle is most conveniently defined as the exponential of a curvature of a connection. It is well known that its cohomology class does not depend on the particular connection chosen. It has been shown by Quillen that a connection may be perturbed by an endomorphism of the vector bundle, such as a symbol of some elliptic differential operator. This point of view, as we intend to show, allows one to relate Chern character to a noncommutative sibling formulated by Connes and Moscovici.     

Homoclinic orbits for a nonperiodic Hamiltonian system

In this paper we prove the existence and multiplicity of homoclinic orbits for first order Hamiltonian systems of the form

     

Global Euler obstruction and polar invariants

Let be a purely dimensional, complex algebraic singular space. We define a global Euler obstruction Eu(Y) which extends the Euler-Poincaré characteristic in case of a nonsingular Y. Using Lefschetz pencils, we express Eu(Y) as alternating sum of global polar invariants. Partially supported by CNRS-CONACYT (12409) Cooperation Program. The first and third named authors partially supported by CONACYT grant G36357-E and DGPA (UNAM) grant IN 101 401.     

Differentiable and algebroid cohomology,Van Est isomorphisms,and characteristic classes

In the first section we discuss Morita invariance of differentiable/algebroid cohomology.In the second section we extend the Van Est isomorphism to groupoids. As a first application we clarify the connection between differentiable and algebroid cohomology (proved in degree 1, and conjectured in degree 2 by Weinstein-Xu [50]). As a second application we extend Van Ests argument for the integrability of Lie algebras. Applied to Poisson manifolds, this immediately implies the integrability criterion of Hector-Dazord [14].In the third section we describe the relevant characteristic classes of representations, living in algebroid cohomology, as well as their relation to the Van Est map. This extends Evens-Lu-Weinsteins characteristic class $\theta_{L}$ [20] (hence, in particular, the modular class of Poisson manifolds), and also the classical characteristic classes of flat vector bundles [2, 30].In the last section we describe applications to Poisson geometry.     

Submersions on open symplectic manifolds

Let M be an open manifold with a symplectic form Ω, and N a manifold with dimN<dimM. We prove that submersions with symplectic fibres satisfy the h-principle. Such submersions define Dirac manifold structures on the given manifold. As an application to this result we show that CPn?CPk−1 admits a submersion into R2(2k−n) with symplectic fibres for n/2<k?n.     

Modification systems and integration in their Chow groups

We introduce a notion of integration on the category of proper birational maps to a given variety X, with value in an associated Chow group. Applications include new birational invariants; comparison results for Chern classes and numbers of nonsingular birational varieties; ‘stringy’ Chern classes of singular varieties; and a zeta function specializing to the topological zeta function. In its simplest manifestation, the integral gives a new expression for Chern–Schwartz–MacPherson classes of possibly singular varieties, placing them into a context in which a ‘change-of-variable’ formula holds.     

The Gauss-Bonnet theorem for vector bundles

We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M. This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when M is a Riemannian manifold and E is the tangent bundle of M endowed with the Levi-Civita connection. The proof is based on an explicit geometric construction of the Thom class for 2-plane bundles. Dedicated to the memory of Philip Bell Research partially supported by NSF grant DMS-9703852.     

Growth functions for semi-regular tilings of the hyperbolic plane

This paper studies the growth function, with respect to the generating set of edge identifications, of a surface group with fundamental domainD in the hyperbolic plane ann-gon whose angles alternate between /p and /q. The possibilities ofn,p andq for which a torsion-free surface group can have such a fundamental polygon are classified, and the growth functions are computed. Conditions are given for which the denominator of the growth function is a product of cyclotomic polynomials and a Salem polynomial.This work was supported in part by NSF Research Grants.     

The Koszul homomorphism for a pair of Lie algebras in the theory of exotic characteristic classes of Lie algebroids

We examine fundamental properties of the universal exotic characteristic homomorphism in the category of Lie algebroids, introduced by the authors in Balcerzak and Kubarski (2012) [4]. The properties under study include: (a) functorial properties with respect to arbitrary morphisms of Lie algebroids, (b) homotopy properties, (c) relationships with the Koszul homomorphism for a pair of isotropy Lie algebras, (d) conditions under which the universal exotic characteristic homomorphism is a monomorphism.     

An α-approximation theorem for Q∞-manifolds

Given a Q^∞-manifold M and an open cover α of M, there is an open cover β of M such that every β-equivalence from a Q^∞-manifold N to M is α-close to a homeomorphism. Consequently, every Q^∞-deficient subset in a C^∞-manifold M is strongly negligible in M.     

Multiple brake orbits in bounded convex symmetric domains

In this paper, we prove that there exist at least two geometrically distinct brake orbits in every bounded convex symmetric domain in .     

Topology of manifolds modeled on countable direct limits of Menger compacta

We construct n-dimensional counterparts of manifolds modeled on the space ?² equipped by the bounded weak topology (-manifolds). For -manifolds we prove the characterization, triangulation and classification theorems. In addition, a universal map of onto Q^∞ (the countable direct limit of Hilbert cubes and Z-embeddings) is constructed and characterized.     

Local structure of Koszul-Vinberg and of Lie algebroids

In this short note we continue our study of Koszul-Vinberg algebroids which form a subcategory of the category of Lie algebroids, and which appear naturally in the study of affine structures, affine and transversally affine foliations [N. Nguiffo Boyom, R. Wolak, J. Geom. Phys. 42 (2002) 307-317]. We prove a local decomposition theorem for KV-algebroids. Using the notion of KV-algebroids we introduce a new class of singular foliations: affine singular foliations. In the last section we study the holonomy of these foliations and prove a stability theorem.