Classes caractéristiques d''une π-algèbre et suite spectrale en K-théorie bivariante

Résumé Pour tout groupe discret et pour toute -algèbre D, la C ^*-algèbre D(E) (dont la définition exacte est donnée dans la section 4) est la version équivariante de la C ^*-algèbre C(B, D) des fonctions continues sur B, le classifiant du groupe, à valeurs dans D et qui s'annulent à l'infini. Si D désigne une autre -algèbre, nous définissons une suite spectrale en K-théorie bivariante dont les premiers termes sont donnés par les groupes H ^p (B, KK(D, D)) et qui converge (lorsque B est de dimension finie) vers KK(B; D(E), D(E)). Cette suite spectrale généralise celle de Kasparov mais est obtenue de manière différente: en étendant la définition des quasihomomorphismes aux C(X)-algèbres (X est une espace topologique localement compact), on a recours à des méthodes homotopiques telles les décompositions de Postnikov et le calcul des groupes d'homotopie des espaces d'équivalences d'homotopie. Sous certaines hypothèses, ces mÊmes constructions nous permettent de définir, pour toute -algèbre D, une obstruction, appelée classe secondaire de la -algèbre D, qui détermine la différentielle d ₂ de la suite spectrale de Kasparov.

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For all discrete group and all -algebra D, the C ⁺-algebra D(E) (whose exact definition is given in Section 4) is the equivariant version of the C ^*-algebra C(B, D) of continuous functions from B (the classifiant of the group) to D, vanishing at infinity. If D is another -algebra, we define a spectral sequence in bivariant K-theory whose first terms are given by the groups H ^p (B, KK(D, D)) and which converges (if B of finite dimension) to KK(B; D(E), D(E)). This spectral sequence generalises the spectral sequence given by Kasparov but it is obtained in a quite different way: by extending the definition of quasihomomorphisms to the C(X)-algebras (where X is a locally compact topological space), we use homotopical methods, like Postnikov decompositions and the calculus of homotopy groups of spaces of homotopy equivalences. Furthermore, under certain hypotheses, with these constructions, we define an obstruction, called the secondary class of the -algebra D, which determines the differential d ₂ of the Kasparov spectral sequence.

     

Nielsensche Realisierungssätze für Seifertfaserungen

Let M be a closed aspherical manifold and A a finite subgroup of the outer automorphism group Out ₁M of ₁M. A necessary (and in many cases also sufficient) condition for realising A by the induced action of an isomorphic group of homeomorphisms of M is the existence of an extension 1₁MEA1 to the abstract kernel (A,₁M, AOut ₁M). If the center of ₁M is nontrivial, this condition need not be fulfilled ([14]). We showed in [25] however that one can always find a surjection BA of a finite group B with abelian kernel such that there exists an extension to the abstract kernel (B,₁M,BAOut₁M), and one can try to realize B instead of A. The main result of the present paper is a characterisation of all such groups B (for a given A) which can be realized by a group of homeomorphisms. The class of manifolds considered here consists of certain Seifert fiber spaces in arbitrary dimensions but the main result is purely algebraic and can be applied to other classes of manifolds, for example to flat Riemannian manifolds.     

Nonlinear instability of superposed magnetic fluids in the presence of an oblique magnetic field

The nonlinear evolution of interfacial waves separating two magnetic fluids subjected to an oblique magnetic field is studied in two dimensions, with the use of the method of multiple scales. It is shown that the evolution of the envelope is governed by two partial differential equations. These equations can be combined to yield two alternate Schrödinger equations with cubic nonlinearity; one of them leads to the determination of the cutoff wave number separating stable from unstable deformations while the other Schrödinger equation is used to analyze the stability of the system. The stability of the system is discussed both theoretically and computationally, and the stability diagrams are obtained. It is found in the linear theory that the oblique magnetic field has a stabilizing influence if 0 ₁ + ₂ < /2, or 3/2 < ₁ + ₂ 2 and a destabilizing influence if /2 < ₁ + ₂ < 3/2, where 0 _j , (j=1, 2) and , is the angle between the field and the horizontal axis.In the nonlinear theory, the stability analysis reveals that there exist different regions of stability and instability. It is reported that the oblique magnetic field plays a dual role in the stability criterion and the angles ₁ and ₂ play a distinctive role in this analysis besides the effect of the variation of the magnetic permeabilities.     

On translations of double rays in graphs

Let be an infinite graph, let be a double ray in , and letd andd denote the distance functions in and in , respectively. One calls anaxis ifd(x,y)=d (x,y) and aquasi-axis if lim infd(x,y)/d (x,y)>0 asx, y range over the vertex set of andd (x,y). The present paper brings together in greater generality results of R. Halin concerning invariance of double rays under the action of translations (i.e., graph automorphisms all of whose vertex-orbits are infinite) and results of M. E. Watkins concerning existence of axes in locally finite graphs. It is shown that if is a translation whose directionD() is a thin end, then there exists an axis inD() andD(^–1) invariant under ^r for somer not exceeding the maximum number of disjoint rays inD().The thinness ofD() is necessary. Further results give necessary conditions and sufficient conditions for a translation to leave invariant a quasi-axis.     

More projective geometry associated with unramified double covers of curves of genus four

Sunto Sia : YY un rivestimento doppio non diramato di una curva di genere quattro definita su C e a moduli generali. Sia il punto di 2-divisione associate a . In questa nota si studia il sistema 2 delle quardriche di contatto al modello canonico di Y, associato al dato rivestimento.e si esplicita una biezione tra l'insieme delle theta-caratteristiche dispari di Y che diffeiscono per e l'insieme dell theta caratteristiche dispari dell curve di genere tre la cui Jacobiana e isomorfa (come v.a.p.p.) alla varieta di Prym P(f Y Y)     

1-formes fermées singulières et groupe fondamental

Summary We study the influence of the ₁ of a closed manifoldM ⁿ (n3) on the foliations ofM defined by closed differential 1-forms with Morse singularities (of index 0,n). Every nonexact form is cohomologous to a weakly complete one, that is one whose leaf space is of the same type as that of a nonsingular form. Generically, a form has compact leaves or is weakly complete. If ₁ M has no quotient isomorphic to *, then every nonexact form onM is weakly complete. We also say a form is complete if every path inM is homotopic to either a path transverse to or a path contained in a leaf of . Completeness of depends only on its de R ham cohomology class. The set of complete cohomology classes depends only on ₁ M and is related to finitely generated normal subgroups of ₁ M with quotient . If ₁ M is nilpotent (or even polycyclic), every nonexact form onM is complete. On irreducible 3-manifolds, a form is complete iff it is cohomologous to a nonsingular one.     

Finite presentability of Steinberg groups over group rings

LetA be a finitely generated commutative -algebra with Krull dimensiond, and let be an arbitrary finite group. It is proved that the Steinberg groupSt _n (A) is finitely presented whenevern4. If, in addition,nd+3, andK ₁ (A) andK ₂ (A) are finitely generated, thenE _n (A) andGL _n (A) are finitely presented.The Project supported by National Natural Science Foundation of China.     

The translation planes of order q2 that admit SL(2,q) as a collineation group. II. Odd order

Let be a translation plane of odd order q², where q=p^r and p is a prime. If admits SL(2,q) (or PSL(2,q)) as a collineation group then is a Desarguesian, Hall, or Hering plane, or one of two Walker planes of order 25.Partially supported by grants from the National Science Foundation.     

Generating alternating permutations lexicographically

A permutation _{1 2} ... _n is alternating if ₁<₂>₃<₄ .... We present a constant average-time algorithm for generating all alternating permutations in lexicographic order. Ranking and unranking algorithms are also derived.Research supported by the Natural Sciences and Engineering Research Council of Canada under grant A3379.     

ApproximatingL 2-invariants by their finite-dimensional analogues

LetX be a finite connectedCW-complex. Suppose that its fundamental group is residually finite, i.e. there is a nested sequence ... _m + 1 _m ... of in normal subgroups of finite index whose intersection is trivial. Then we show that thep-thL ²-Betti number ofX is the limit of the sequenceb _p(X_m)/[: _m ] whereb _p(X_m) is the (ordinary)p-th Betti number of the finite covering ofX associated with _m .     

Limites dégénérées de séries discrètes, formes automorphes et variétés de Griffiths–Schmid: le cas du groupe U (2, 1)

Let G be an inner anisotropic form of an unitary group of 3 variables over Q, such that G_RU(2,1), and be an automorphic representation of G(A) whose archimedean component is a degenerate limit of discrete series; such a never occurs in the cohomology (coherent or étale) of a Shimura variety. We show that however it does appear in the coherent cohomology of some line bundle over an associated Griffiths-Schmid variety. Moreover we study cup products between such cohomology classes and some other automorphic cohomology classes and we prove some non-vanishing results.     

Relative komplexe quotienten

Let XS be a holomorphic map, and let RX×SX be an equivalence relation. The restriction of R to the fibre ^–1(S) is denoted by R_s. The quotient X/R is called a relative complex quotient, if the quotient map XX/R is holomorphic over S. Two cases are studied: (C) All fibres of are locally R_s-separable (relative Cartan quotient); (R) All fibres of are holomorphically convex, and R_s is given by tke holomorphic functions on ^–1 (s) (relative Remmert quotient).     

Sui piani di André generalizzati privi di orbite di lunghezza finita sulla retta impropria

This paper presents an example of an infinite generalized André plane with the following properties: (1) has dimension eight over its kernel (hence is neither desarguesian nor a Hall plane); (2) the full collineation group of has no orbit of finite length on the line at infinity.

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Lavoro eseguito nell'ambito delle attività del G.N.S.A.G.A. del C.N.R.     

Subharmonic solutions of nonlinear wave equations onS n

In this paper we study a nonlinear wave equation on Sⁿ with a nonlinear term 2-periodic in t. More precisely we give a Birkoff-Lewis type result of existence of periodic solutions localized near an equilibrium point, that is we search for distinct 2k-periodic solutions (subharmonics) of the wave equation, k, which have small amplitude and minimal period 2k.Work supported by M.U.R.S.T. (Research funds 40% and 60%).     

Generalization of istratescu's theorem on contractive mappings in metric spaces

In this paper we consider mappings T:XX of metric spaces, satisfying the condition: , where is some right semicontinuous function. We prove that if is a nondecreasing function, ()< for >0, –() as ,, then the map T has a fixed point and for any pointxX. Interesting examples are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol, 83, pp. 73–82, 1979.     

Transposition generation of alternating permutations

A permutation _{1 2}... _n is alternating if ₁< ₂> ₃< ₄.... Alternating permutations are counted by the Euler numbers. Here we show that alternating permutations can be listed so that successive permutations differ by a transposition, ifn is odd. Extensions and open problems are mentioned.Research supported by the Natural Sciences and Engineering Research Council of Canada under grant A3379.     

An initial-boundary value problem for the sine-Gordon equation in laboratory coordinates

We consider the sine-Gordon equation in laboratory coordinates with both x and t in [0, ). We assume that u(x, 0), u_t(x, 0), u(0, t) are given, and that they satisfy u(x, 0)2q, u_t(x, 0)0, for large x, u(0, t)2p for large t, where q, p are integers. We also assume that u_x(x, 0), u_t(x, 0), u_t(0, t), u(0, t)-2p, u(x, 0)-2q L₂. We show that the solution of this initial-boundary value problem can be reduced to solving a linear integral equation which is always solvable. The asymptotic analysis of this integral equation for large t shows how the boundary conditions can generate solitons.The authors dedicate this paper to the memory of M. C. PolivanovDepartment of Mathematics and Computer Science; Institute for Nonlinear Studies, Clarkson University, Postdam, New York. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 3, pp. 387–403, September, 1992.     

Range of the posterior probability of an interval for priors with unimodality preserving contaminations

Range of the posterior probability of an interval over the -contamination class ={=(1–)₀+q:qQ} is derived. Here, ₀ is the elicited prior which is assumed unimodal, is the amount of uncertainty in ₀, andQ is the set of all probability densitiesq for which =(1–)₀+q is unimodal with the same mode as that of ₀. We show that the sup (resp. inf) of the posterior probability of an interval is attained by a prior which is equal to (1–)₀ except in one interval (resp. two disjoint intervals) where it is constant.