The tangential end fibration of an aspherical Poincaré complex

We construct a sphere fibration over a finite aspherical Poincaré complex X, which we call the tangential end fibration, under the condition that the universal cover of X is forward tame and simply connected at infinity. We show that it is tangent to X if the formal dimension of X is even or, when the formal dimension is odd, if the diagonal X→X×X admits a Poincaré embedding structure.     

Infinitesimal Liouville Distributions for Teichmuller space

We consider an arbitrary Riemann surface X, possibly of infinitehyperbolic area. The Liouville measure of the hyperbolic metricdefines a measure on the space of geodesics of the universal covering of X. As we vary the Riemann surface structure,this gives an embedding from the Teichmuller space of X intothe Fréchet space of Hölder distributions on G. We show that the embedding is continuouslydifferentiable. In particular, we obtain an explicit integralrepresentation of the tangent map. 2000 Mathematics SubjectClassification 30F60, 32G15 (primary), 46F99 (secondary).     

Cohomology Actions and Centralisers in Unitary Reflection Groups

In an earlier work, the second author proved a general formulafor the equivariant Poincaré polynomial of a linear transformationg which normalises a unitary reflection group G, acting on thecohomology of the corresponding hyperplane complement. Thisformula involves a certain function (called a Z-function below)on the centraliser CG(g), which was proved to exist only incertain cases, for example, when g is a reflection, or is G-regular,or when the centraliser is cyclic. In this work we prove theexistence of Z-functions in full generality. Applications includereduction and product formulae for the equivariant Poincarépolynomials. The method is to study the poset L(C_G(g)) of subspaceswhich are fixed points of elements of C_G(g). We show that thisposet has Euler characteristic 1, which is the key propertyrequired for the definition of a Z-function. The fact aboutthe Euler characteristic in turn follows from the ‘join-atom’property of L(C_G(g)), which asserts that if [X₁,..., X_k} isany set of elements of L(C_G(g)) which are maximal (set theoretically)then their setwise intersection lies in L(C_G(g)). 2000 Mathematical Subject Classification:primary 14R20, 55R80; secondary 20C33, 20G40.     

An algebraic loop theorem and the decomposition of PD3-pairs

Let (Y, X) denote a three-dimensional Poincaré pair (PD³-pair).By the work of Eckmann, Müller and Linnell we may suppose,up to a homotopy equivalence, that the boundary X is a closed2-manifold. We show that if a component of X fails to be ₁-injectivein Y, then there is an essential simple loop in X which is nullhomotopicin Y. It follows that there is a finite process of attaching2-disks along essential simple loops on X, and filling sphericalcomponents of X, which transforms (Y, X) into a PD³-pair (Y',X') with aspherical incompressible boundary X' and such that₁(Y) = ₁(Y'). The PD³-pair (Y', X') then admits a canonicaldecomposition as a connected sum of a finite number of asphericalPD³-pairs with incompressible boundary, together with a PD³-pairhaving virtually free (possibly finite) fundamental group andboundary a (possibly empty) disjoint union of projective planes.     

Poincare Series of Multi-Filtered Algebras and Partitivity

It is proved that if an algebra R over a field can be endowedwith a pointed and finite-dimensional Nⁿ-filtration such thatthe associated Nⁿ-graded algebra T is semi-commutative, thenR is left and right finitely partitive. In order to do this,a multi-variable Poincaré series for every finitely generatedgraded T-module is considered and it is shown that this Poincaréseries is a rational function. The methods apply to some iteratedOre extensions such as quantum matrices and quantum Weyl algebrasas well as to the quantized enveloping algebra of sl(+1).     

Poincare Series and Zeta Function for an Irreducible Plane Curve Singularity

The Poincaré series of an irreducible plane curve singularityequals the -function of its monodromy, by a result of Campillo,Delgado and Gusein-Zade. This fact is derived in this paperfrom a formula of Ebeling and Gusein-Zade, relating the Poincaréseries of a quasi-homogeneous complete intersection singularityto the Saito dual of a product of -functions. 2000 MathematicsSubject Classification 32S40 (primary), 14B05 (secondary).     

Classification Des Ideaux a Droite De A1(k)

Les études récentes sur les idéaux àdroite de A₁(k), la première algèbre de Weyl surun corps algébriquement clos et de caractéristiquenulle k, nous montrent que : pour tout idéal I 0 àdroite de A₁(k), il existe x Q = frac(A₁(k)), et V V telsque : I = xD(R, V) o V est l'ensemble des sous-espaces primairementdécomposables de k[t] = R, et D(R, V), l'idéalà droite {d A₁(k/d(R V}. Dans cet article nous montreronsprincipalement que: pour tout 0 I idéal à droitede A₁(k, !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),où X_n est la courbe d'algèbre des fonctions régulières: O(X_n = k+tⁿ⁺¹k[t]. La forme des idéaux décriteci-dessus permet de voir dans une hypothèse de Letzteret Makar-Limanov, pour deux courbes algébriques affinesX et X' on a : D(XD(X') co dim D(X = co dim D(X'). Recent studies on right ideals of the first Weyl algebra A₁(k)over an algebraic closed field k with characteristic zero showthat: for each right ideal I 0 of A₁(k), there exist x Q =fracA₁(k)) and a primary decomposable sub-space V of k[t] suchthat I=xD(R,V), where D(R,V) : = {d A₁(k)/d(R) V} is a rightideal of A₁(k). In this paper, we show that for all right idealsI 0 of A₁(k), !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),where X_n denotes the affine algebraic curve with ring of regularfunctions O(X_n=k+tⁿ⁺¹k[t]. With ideals as described above, onecan easily see, under a hypothesis given by Letzter and Makar-Limanov,that for two affine algebraic curves X and X', D(X)D(X') codim D(X) = co dim D(X'). 2000 Mathematics Subject Classification16S32.     

Rangs Stables De Certaines Extensions

M. A. Rieffel [24] a introduit le rang stable topologique (tsr),pour généraliser aux C*-algèbres, le conceptde dimension de recouvrement pour les espaces compacts, affirmantainsi le principe selon lequel une C*-algèbre est ‘unespace localement compact non commutatif’. II a montréque l'on a tsr ((A)) = [dim (Â)] + 1, pour toute C*-algèbre commutative A etque trs (B/J) tsr (B), pour toute C*-algèbre B et pourtout idéal bilatère fermé J dans B (généralisantle fait que, si X est un espace compact et F un sous-ensemblefermé dans X, alors on a dim (F) dim (X), oùdim(X) est la dimension de recouvrement de X [19]). D'autrepart, le rang stable topologique peut être utilisépour obtenir des théorèmes de ‘cancellation’pour les modules projectifs, comme ceci est fait dans [25, 2].Un peu plus tard, R. H. Herman et L. N. Vaserstein [14] ontmontré que pour toute C*-algèbre unitaire A, lerang stable topologique de A et le rang stable de Basse de Acoincident, done, pour toute C*-algèbre unitaire A, onnote sr(A) cette valeur commune appelée rang stable deA. Les C*-algèbres unitaires de rang stable 1 ont étéétudiée géométriquement par M. Rørdam[27], il a montré que l'on a sr(A) = 1 si et seulementsi l'enveloppe convexe des unitaires de A est égale àla boule unité fermé de A. D'autre part, Rieffel[24] avait introduit le rang stable connexe (csr) d'une C*-algèbre,sur lequel V. Nistor [18] a publié un article trésintéressant. Mon travail dans ce papier consiste àcompléter certains travaux déjà entreprisdans les articles qui sont cités ci-dessus.     

Higher string topology on general spaces

In this paper, I give a generalized analogue of the string topologyresults of Chas and Sullivan, and of Cohen and Jones. For afinite simplicial complex X and k 1, I construct a spectrumMaps(S^k, X)^S(X), which is obtained by taking a generalizationof the Spivak bundle on X (which however is not a stable spherebundle unless X is a Poincaré space), pulling back toMaps(S^k, X) and quotienting out the section at infinity. I showthat the corresponding chain complex is naturally homotopy equivalentto an algebra over the (k + 1)-dimensional unframed little diskoperad C_{k + 1}. I also prove a conjecture of Kontsevich, whichstates that the Quillen cohomology of a based C_k-algebra (inthe category of chain complexes) is equivalent to a shift ofits Hochschild cohomology, as well as prove that the operadC_*C_k is Koszul-dual to itself up to a shift in the derived category.This gives one a natural notion of (derived) Koszul dual C_*C_k-algebras.I show that the cochain complex of X and the chain complex of^k X are Koszul dual to each other as C_*C_k-algebras, and thatthe chain complex of Maps(S^k, X)^S(X) is naturally equivalentto their (equivalent) Hochschild cohomology in the categoryof C_* C_k-algebras. 2000 Mathematics Subject Classification 55P48(primary), 16E40, 55N45, 18D50 (secondary).     

VECTOR FIELDS IN R2 WITH MAXIMAL INDEX

We consider the method of Poincaré to investigate thelocal index of vector fields in the plane. If m is the degreeof the first non-zero jet, X_m, of the vector field X at an isolatedzero, we explore the geometry of the pencil generated by thecoordinate functions of X_m when the absolute value of the indexof X, |ind (X)|, is m. We also find necessary and sufficientconditions for |ind (X)| to be m.     

Rank Properties of Endomorphisms of Infinite Partially Ordered Sets

The relative rank (S : U) of a subsemigroup U of a semigroupS is the minimum size of a set V S such that U together withV generates the whole of S. As a consequence of a result ofSierpiski, it follows that for U T_X, the monoid of all self-mapsof an infinite set X, rank(T_X : U) is either 0, 1 or 2, or uncountable.In this paper, the relative ranks rank(T_X : O_X) are considered,where X is a countably infinite partially ordered set and O_Xis the endomorphism monoid of X. We show that rank(T_X : O_X) 2 if and only if either: there exists at least one elementin X which is greater than, or less than, an infinite numberof elements of X; or X has |X| connected components. Four examplesare given of posets where the minimum number of members of T_Xthat need to be adjoined to O_X to form a generating set is,respectively, 0, 1, 2 and uncountable. 2000 Mathematics SubjectClassification 08A35 (primary), 06A07, 20M20 (secondary).     

Hyperbolic Hexagons and Algebraic Curves in Genus 3

One of the consequences of the uniformization theorem of Koebeand Poincaré is that any smooth complex algebraic curveC of genus g > 1 is conformally equivalent to H/G, whereG PSL₂(R) is a Fuchsian group and is naturally endowed witha hyperbolic metric. Conversely, any compact hyperbolic surfaceis isomorphic to an algebraic curve. Hence any curve of genusg > 1 may be described in two ways, either by an equationor by a Fuchsian group.     

A Strengthening of Resolution of Singularities in Characteristic Zero

Let X be a closed subscheme embedded in a scheme W, smooth overa field k of characteristic zero, and let I (X) be the sheafof ideals defining X. Assume that the set of regular pointsof X is dense in X. We prove that there exists a proper, birationalmorphism, : W_r W, obtained as a composition of monoidal transformations,so that if X_r W_r denotes the strict transform of X W then: (1) the morphism : W_r W is an embedded desingularization ofX (as in Hironaka's Theorem); (2) the total transform of I (X) in factors as a product of an invertible sheaf of ideals L supportedon the exceptional locus, and the sheaf of ideals defining thestrict transform of X (that is, . Thus (2) asserts that we can obtain, in a simple manner, theequations defining the desingularization of X. 2000 MathematicalSubject Classification: 14E15.     

Global smooth fibrations of R3 by oriented lines

A smooth fibration of ³ by oriented lines is given by a smoothunit vector field V on ³ all of whose integral curves are straightlines. Such a fibration is said to be nondegenerate if dV vanishesonly in the direction of V. Let be the space of oriented linesof ³ endowed with its canonical pseudo-Riemannian neutral metric.We characterize the nondegenerate smooth fibrations of ³ byoriented lines as the closed (in the relative topology) definiteconnected surfaces in . In particular, local conditions on imply the existence of a global fibration. Besides, for anysuch fibration the base space is diffeomorphic to the open discand the directions of the fibers form an open convex set ofthe two-sphere. We characterize as well, in a similar way, thesmooth (possibly degenerate) fibrations.     

MOTIVIC INVARIANTS OF ARTIN STACKS AND 'STACK FUNCTIONS'

An invariant of quasiprojective -varieties X with values ina commutative ring is motivic if (X) = (Y) + (X\ Y) for Y closedin X, and (X x Y) = (X)(Y). Examples include Euler characteristics and virtual Poincaré and Hodge polynomials. We firstdefine a unique extension ' of to finite type Artin -stacks, which is motivic and satisfies '([X/G]) = (X)/(G) when X is a -variety, G a special -groupacting on X, and [X/G] is the quotient stack. This only worksif (G) is invertible in for all special -groups G, which excludes = as (_m) = 0. But we can extend the construction to get roundthis. Then we develop the theory of stack functions on Artin stacks.These are a universal generalization of constructible functionson Artin stacks. There are several versions of the construction:the basic one , and variants ‘twisted’ by motivic invariants. We associate a -vector space or a -module to each Artin stack , with functorial operations of multiplication, pullbacks * and pushforwards _*under 1-morphisms ;, and so on. They will be important tools in the author's series on ‘Configurationsin abelian categories’.     

Twistors and 3-symmetric spaces

We describe complex twistor spaces over inner 3-symmetric spacesG/H, such that H acts transitively on the fibre. As in the symmetriccase, the complex twistor spaces are flag manifolds G/K, whereK is the centralizer of a torus in G. Moreover, they carry analmost complex structure defined using the horizontal distributionof the normal connection on G/H that coincides with the complexstructure associated to a parabolic subgroup PG if it is integrable.Conversely, starting from a complex flag manifold G/P, thereexists a natural fibration with complex fibres on a 3-symmetricspace, called fibration of degree 3.     

Zel'Manov's Theorem for Primitive Jordan-Banach Algebras

The following result is well known and easy to prove (see [14,Theorem 2.2.6]). Theorem 0. If A is a primitive associative Banach algebra, thenthere exists a Banach space X such that A can be seen as a subalgebraof the Banach algebra BL(X) of all bounded linear operatorson X in such a way that A acts irreducibly on X and the inclusionABL(X) is continuous. In fact, if X is any vector space on which the primitive Banachalgebra A acts faithfully and irreducibly, then X can be convertedin a Banach space in such a way that the requirements in Theorem0 are satisfied and even the inclusion ABL(X) is contractive. Roughly speaking, the aim of this paper is to prove the appropriateJordan variant of Theorem 0.     

Minimum number of ideal generators for a linear center perturbed by homogeneous polynomials

Using the algorithm presented in [J. Giné, X. Santallusia, On the Poincaré-Liapunov constants and the Poincaré series, Appl. Math. (Warsaw) 28 (1) (2001) 17-30] the Poincaré-Liapunov constants are calculated for polynomial systems of the form , , where P_n and Q_n are homogeneous polynomials of degree n. The objective of this work is to calculate the minimum number of ideal generators i.e., the number of functionally independent Poincaré-Liapunov constants, through the study of the highest fine focus order for n=4 and n=5 and compare it with the results that give the conjecture presented in [J. Giné, On the number of algebraically independent Poincaré-Liapunov constants, Appl. Math. Comput. 188 (2) (2007) 1870-1877]. Moreover, the computational problems which appear in the computation of the Poincaré-Liapunov constants and the determination of the number of functionally independent ones are also discussed.     

Sharp Geometric Poincare Inequalities for Vector Fields and Non-Doubling Measures

We derive Sobolev–Poincaré inequalities that estimatethe L^q(d µ) norm of a function on a metric ball when µis an arbitrary Borel measure. The estimate is in terms of theL¹(d ) norm on the ball of a vector field gradient of the function,where d dx is a power of a fractional maximal function of µ.We show that the estimates are sharp in several senses, andwe derive isoperimetric inequalities as corollaries. 1991 MathematicsSubject Classification: 46E35, 42B25.     

Weighted Sobolev Interpolation Inequalities on Certain Domains

We obtain some weighted Sobolev interpolation inequalities oncertain domains that include Lipschitz domains for doublingweights satisfying a weighted Poincaré inequality. Thesegeneralize most results in Gutierrez and Wheeden's paper [20].We also give some applications on Lipschitz domains for weightsof type dist (·, G)⁸, where G .