The Gysin sequence for S3{\mathbb S}^3-actions on manifolds

We construct a Gysin sequence associated to any smooth ${\mathbb S}^3$-action on a smooth manifold.Comment: Accepted for publication in Publicationes Mathematicae Debrecen, scheduled for 2014 Publicationes Mathematicae Debrecen (2014

Top dimensional group of the basic intersection cohomology for singular riemannian foliations

It is known that, for a regular riemannian foliation on a compact manifold, the properties of its basic cohomology (non-vanishing of the top-dimensional group and Poincar\'e Duality) and the tautness of the foliation are closely related. If we consider singular riemannian foliations, there is little or no relation between these properties. We present an example of a singular isometric flow for which the top dimensional basic cohomology group is non-trivial, but its basic cohomology does not satisfy the Poincar\'e Duality property. We recover this property in the basic intersection cohomology. It is not by chance that the top dimensional basic intersection cohomology groups of the example are isomorphic to either 0 or $\mathbb{R}$. We prove in this Note that this holds for any singular riemannian foliation of a compact connected manifold. As a Corollary, we get that the tautness of the regular stratum of the singular riemannian foliation can be detected by the basic intersection cohomology.Comment: 11 pages. Accepted for publication in the Bulletin of the Polish Academy of Science

Diversos sentidos de la palabra dolor

Los campos semánticos del dolor son principalmente dos: dolor físico, dolor moral. Mientras que el primero sirve como testigo de alguna disfunción corporal, y por eso inicialmente se puede considerar que es algo positivo, el segundo es más complicado de asumir: ¿qué sentido puede tener el sufrimiento moral de una persona?, ¿para qué le sirve? ¿qué lo causa? A lo largo de este artículo se intenta hacer una descripción fenomenológica de los sentidos que adquiere en el lenguaje ordinario el dolor, tratando de desentrañar qué función puede tener para la vida humana esa oscura experiencia

Tautness for riemannian foliations on non-compact manifolds

For a riemannian foliation $\mathcal{F}$ on a closed manifold $M$, it is known that $\mathcal{F}$ is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form $\kappa_\mu$ (relatively to a suitable riemannian metric $\mu$) is zero. In the transversally orientable case, tautness is equivalent to the non-vanishing of the top basic cohomology group $H^{^{n}}(M/\mathcal{F})$, where n = \codim \mathcal{F}. By the Poincar\'e Duality, this last condition is equivalent to the non-vanishing of the basic twisted cohomology group $H^{^{0}}_{_{\kappa_\mu}}(M/\mathcal{F})$, when $M$ is oriented. When $M$ is not compact, the tautness class is not even defined in general. In this work, we recover the previous study and results for a particular case of riemannian foliations on non compact manifolds: the regular part of a singular riemannian foliation on a compact manifold (CERF).Comment: 18 page