Une famille remarquable de suites recurrentes lineaires

Letu(n) be a recurrent sequence of rational integers, i.e.,u(n+s)+a _s–1 u(n+s–1)+...+a ₀ u(n)=0,n0,a _i,i=0,...,s–1. The polynomialP(x)=x ^s +a _s–1x^s +...+a ₀ is the companion or the characteristic polynomial of the recurrence. It is known that if none of the ratios of the roots ofP is a root of unity, then the setA={n,u(n)=0} is finite. A recent result of F. Beukers shows that ifs=3, then the setA has at most 6 elements and there exists, up to trivial transformations, only one recurrence of order 3 with 6 zeros, found by J. Berstel. In this paper, we construct for eachs, s2 a recurrent sequence of orders, with at leasts ²/2+s/2–1 zeroes, which generalize Berstel's sequence.

     

A note on the volume flux group of four-manifolds

We show that closed orientable smooth four-manifolds with non-trivial volume flux group and fundamental group of subexponential growth type are finitely covered by a manifold homeomorphic to S³×S¹, S²×T² or a nil-manifold. We also show that if a compact complex surface has non-trivial volume flux group then it has zero minimal volume.     

Homotopy decomposition of a group of symplectomorphisms of S×S

We continue the analysis started by Abreu, McDuff and Anjos of the topology of the group of symplectomorphisms of S²×S² when the ratio of the area of the two spheres lies in the interval (1,2]. We express the group, up to homotopy, as the pushout (or amalgam) of certain of its compact Lie subgroups. We use this to compute the homotopy type of the classifying space of the group of symplectomorphisms and the corresponding ring of characteristic classes for symplectic fibrations.     

On the diffeomorphism groups of elliptic surfaces

In this paper we determine for relatively minimal elliptic surfaces with positive Euler number the image of the natural representation of the group of orientation preserving self-diffeomorphisms on , the second homology group reduced modulo torsion. To this end we construct as many embedded spheres of square such that an isometry not induced from any combination of reflections at such spheres or from ‘complex conjugation’ can be shown not to be induced from some diffeomorphism at all. This is done with the help of Seiberg-Witten-invariants. Received: 18 December 1995 / Revised version: 27 January 1997     

Radford's biproduct for quasi-Hopf algebras and bosonization

Let B be a braided Hopf algebra (with bijective antipode) in the category of left Yetter-Drinfeld modules over a quasi-Hopf algebra H. As in the case of Hopf algebras (J. Algebra 92 (1985) 322), the smash product B#H defined in (Comm. Algebra 28(2) (2000) 631) and a kind of smash coproduct afford a quasi-Hopf algebra structure on B⊗H. Using this, we obtain the structure of quasi-Hopf algebras with a projection. Further we will use this biproduct to describe the Majid bosonization (J. Algebra 163 (1994) 165) for quasi-Hopf algebras.     

Geometric diffeomorphism finiteness in low dimensions and homotopy group finiteness

The main results of this note consist in the following two geometric finiteness theorems for diffeomorphism types and homotopy groups of closed simply connected manifolds: 1. For any given numbers C and D the class of closed smooth simply connected manifolds of dimension which admit Riemannian metrics with sectional curvature bounded in absolute value by $\vert K \vert\le C$ and diameter bounded from above by D contains at most finitely many diffeomorphism types. In each dimension there exist counterexamples to the preceding statement. 2. For any given numbers C and D and any dimension m there exist for each natural number up to isomorphism always at most finitely many groups which can occur as the k-th homotopy group of a closed smooth simply connected m-manifold which admits a metric with sectional curvature and diameter . Received: 21 August 1999 / Accepted: 20 April 2001 / Published online: 19 October 2001     

The group of parenthesized braids

We investigate a group B_• that includes Artin's braid group B_∞ and Thompson's group F. The elements of B_• are represented by braids diagrams in which the distances between the strands are not uniform and, besides the usual crossing generators, new rescaling operators shrink or stretch the distances between the strands. We prove that B_• is a group of fractions, that it is orderable, admits a nontrivial self-distributive structure, i.e., one involving the law x(yz)=(xy)(xz), it embeds in the mapping class group of a sphere with a Cantor set of punctures, and that Artin's representation of B_∞ into the automorphisms of a free group extends to B_•.     

Localization and residue of the Bott class

The Bott class of transversely holomorphic foliations is studied. We first introduce a formula which relates the Bott class and the Godbillon-Vey class. Then a ‘localizable part’ of the Bott class is defined. It is indeed localizable and written in terms of the Godbillon measure studied by Heitsch and Hurder. The above-mentioned formula is reviewed in terms of localizable parts. Finally, complex codimension-one foliations are considered. A version of residue is introduced and it is shown that the Bott class is ‘localized’ near the Julia set in the sense of Ghys-Gomez-Mont-Saludes. Some examples of calculation of the residue are presented.     

Determining whether a finite group of outer automorphisms of a free group is geometric

In this article we give necessary and sufficient conditions for a given finite group of outer automorphisms to be induced by the action of a group of orientation-preserving homeomorphisms on the fundamental group of a punctured surface. When the group is abelian, necessary and sufficient conditions can also be given in the absence of orientability assumptions. These properties are formulated in terms of the finite automorphism groups which project into the given outer automorphism group: each non-trivial automorphism in any such group can fix at most a cyclic subgroup of the fundamental group.     

Nonsmoothable group actions on elliptic surfaces

Let G be a cyclic group of order 3, 5 or 7, and X=E(n) be the relatively minimal elliptic surface with rational base. In this paper, we prove that under certain conditions on n, there exists a locally linear G-action on X which is nonsmoothable with respect to infinitely many smooth structures on X. This extends the main result of [X. Liu, N. Nakamura, Pseudofree Z/3-actions on K3 surfaces, Proc. Amer. Math. Soc. 135 (3) (2007) 903-910].     

The mod 2 cohomology of the mapping class group of the Klein bottle with marked points

The purpose of this article is to compute the mod 2 cohomology of ${\mathrm{\Gamma }}^q(\mathbb{K})$, the mapping class group of the Klein bottle with q marked points. We provide a concrete construction of Eilenberg–MacLane spaces $X_q=K({\mathrm{\Gamma }}^q(\mathbb{K}),1)$ and fiber bundles $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q\to X_q\to B({\mathbb{Z}}_2\times O(2))$, where $F_q(\mathbb{K})/{\mathrm{\Sigma }}_q$ denotes the configuration space of unordered q-tuples of distinct points in $\mathbb{K}$ and $B({\mathbb{Z}}_2\times O(2))$ is the classifying space of the group ${\mathbb{Z}}_2\times O(2)$. Moreover, we show the mod 2 Serre spectral sequence of the bundle above collapses.     

On the arc component of a locally compact Abelian group

For a topological group G, we denote by G _a the arc component of the neutral element and by the character group of G, i.e. the group of all continuous homomorphisms from G into T. We prove the following theorem: Let G be a connected locally compact abelian group and let be the embedding. Then is a topological isomorphism. In particular, the character group of the arc component of a compact abelian group is discrete. Some conclusions will be drawn.     

Compactifications of the homeomorphism group of a graph

Let Γ be a countable locally finite graph and let H(Γ) and H₊(Γ) denote the homeomorphism group of Γ with the compact-open topology and its identity component. These groups can be embedded into the space of all closed sets of Γ×Γ with the Fell topology, which is compact. Taking closure, we have natural compactifications and . In this paper, we completely determine the topological type of the pair and give a necessary and sufficient condition for this pair to be a (Q,s)-manifold. The pair is also considered for simple examples, and in particular, we find that has homotopy type of RP³. In this investigation we point out a certain inaccuracy in Sakai-Uehara's preceding results on for finite graphs Γ.     

The missing boundary problem for smooth manifolds of dimension greater than or equal to six

Let M be a non-compact differentiable manifold of dimension ?6. Suppose both M and ?M are 1-ended spaces. We give necessary and sufficient conditions for M to be diffeomorphic to the complement of a compact subset of the boundary of a compact manifold. There are four conditions: two geometric conditions and two algebraic obstructions. We give examples to show that these obstructions are not always trivial. In particular, an example of a manifold is constructed which does not have a completion but any tubular neighborhood of codimension ?3 has a completion. We also classify the different ways to complete a given manifold.     

A separable manifold failing to have the homotopy type of a CW-complex

We show that (a variation of) the Prüfer surface, which is an example of a separable non-metrizable 2-manifold, does not have the homotopy type of a CW-complex. Received: 6 February 2007 Revised: 21 August 2007     

On the index of tricyclic graphs with perfect matchings

Let T(2k) be the set of all tricyclic graphs on 2k(k?2) vertices with perfect matchings. In this paper, we discuss some properties of the connected graphs with perfect matchings, and then determine graphs with the largest index in T(2k).     

Note on a Theorem of Munkres

We prove that given a Riemannian manifold with boundary, having a finite number of compact boundary components, any fat triangulation of the boundary can be extended to the whole manifold. We also show that this result extends to manifolds and to embedded PL manifolds of dimensions 2, 3 and 4. We employ these results to prove that manifolds of the types above admit quasimeromorphic mappings onto As an application we prove the existence of G-automorphic quasimeromorphic mappings, where G is a Kleinian group acting on Dedicated to the memory of Robert BrooksThis paper represents part of the authors Ph.D. thesis written under the supervision of Prof. Uri Srebro.