Three-manifolds of constant vector curvature one

A Riemannian manifold has $\mathrm{CVC}(?)$ if its sectional curvatures satisfy $\sec \le \epsilon$ or $\sec \ge \epsilon$ pointwise, and if every tangent vector lies in a tangent plane of curvature ε. We present a construction of an infinite-dimensional family of compact $\mathrm{CVC}(1)$ three-manifolds.     

Generalized local cohomology and regularity of Ext modules

Let R be a Noetherian standard graded ring, and M and N two finitely generated graded R-modules. We introduce ${\mathrm{reg}}_R(M,N)$ by using the notion of generalized local cohomology instead of local cohomology, in the definition of regularity. We prove that ${\mathrm{reg}}_R(M,N)$ is finite in several cases. In the case that the base ring is a field, we show that${\mathrm{reg}}_R(M,N)=\mathrm{reg}(N)?\mathrm{indeg}(M).$ This formula, together with a graded version of duality for generalized local cohomology, gives a formula for the minimum of the initial degrees of some Ext modules (in the case R is Cohen–Macaulay), of which the three usual definitions of regularity are special cases. Bounds for regularity of certain Ext modules are obtained, using the same circle of ideas.     

Contracting an element from a cocircuit

We consider the situation that M and N are 3-connected matroids such that $|E(N)|⩾4$ and $C^1$ is a cocircuit of M with the property that $M/x_0$ has an N-minor for some $x_0\in C^1$. We show that either there is an element $x\in C^1$ such that $\mathrm{si}(M/x)$ or $\mathrm{co}(\mathrm{si}(M/x))$ is 3-connected with an N-minor, or there is a four-element fan of M that contains two elements of $C^1$ and an element x such that $\mathrm{si}(M/x)$ is 3-connected with an N-minor.     

Floer homology for almost Hamiltonian isotopies

Seidel introduced a homomorphism from the fundamental group ${\pi }_1(\mathrm{Ham}(M))$ of the group of Hamiltonian diffeomorphisms of certain compact symplectic manifolds $(M,\omega )$ to a quotient of the automorphism group $\mathrm{Aut}({\mathit{HF}}_1(M,\omega ))$ of the Floer homology ${\mathit{HF}}_1(M,\omega )$. We prove a rigidity property: if two Hamiltonian loops represent the same element in ${\pi }_1(\mathrm{Diff}(M))$, then the image under the Seidel homomorphism of their classes in ${\pi }_1(\mathrm{Ham}(M))$ coincide. The proof consists in showing that Floer homology can be defined by using ‘almost Hamiltonian’ isotopies, i.e. isotopies that are homotopic relatively to endpoints to Hamiltonian isotopies. To cite this article: A. Banyaga, C. Saunders, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Attaching topological spaces to a module (I): Sobriety and spatiality

In this paper we study some frames associated to an R-module M. We define semiprimitive submodules and we prove that they form an spatial frame canonically isomorphic to the topology of $\mathrm{Max}(M)$. We characterize the soberness of $\mathrm{Max}(M)$ in terms of the point space of that frame. Beside of this, we study the regularity of an spatial frame associated to M given by annihilator conditions.