Top-Dimensional Group of the Basic Intersection Cohomology for Singular Riemannian Foliations

José Ignacio Royo Prieto, Martintxo Saralegi-Aranguren, and Robert Wolak. "Top-Dimensional Group of the Basic Intersection Cohomology for Singular Riemannian Foliations." Bulletin of the Polish Academy of Sciences. Mathematics 53.4 (2005): 429-440. <http://eudml.org/doc/280409>.

@article{JoséIgnacioRoyoPrieto2005,
abstract = {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 Poincaré 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 the basic cohomology does not satisfy the Poincaré Duality. However, we recover the Poincaré Duality in the basic intersection cohomology. It is not accidental that the top-dimensional basic intersection cohomology groups of the example are isomorphic to either 0 or ℝ. We prove that this holds for any singular riemannian foliation of a compact connected manifold. As a corollary, we show that the tautness of the regular stratum of the singular riemannian foliation can be detected by the basic intersection cohomology.},
author = {José Ignacio Royo Prieto, Martintxo Saralegi-Aranguren, Robert Wolak},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {singular Riemannian foliation; basic intersection cohomology; basic cohomology; taut foliation},
language = {eng},
number = {4},
pages = {429-440},
title = {Top-Dimensional Group of the Basic Intersection Cohomology for Singular Riemannian Foliations},
url = {http://eudml.org/doc/280409},
volume = {53},
year = {2005},
}

TY - JOUR
AU - José Ignacio Royo Prieto
AU - Martintxo Saralegi-Aranguren
AU - Robert Wolak
TI - Top-Dimensional Group of the Basic Intersection Cohomology for Singular Riemannian Foliations
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2005
VL - 53
IS - 4
SP - 429
EP - 440
AB - 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 Poincaré 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 the basic cohomology does not satisfy the Poincaré Duality. However, we recover the Poincaré Duality in the basic intersection cohomology. It is not accidental that the top-dimensional basic intersection cohomology groups of the example are isomorphic to either 0 or ℝ. We prove that this holds for any singular riemannian foliation of a compact connected manifold. As a corollary, we show that the tautness of the regular stratum of the singular riemannian foliation can be detected by the basic intersection cohomology.
LA - eng
KW - singular Riemannian foliation; basic intersection cohomology; basic cohomology; taut foliation
UR - http://eudml.org/doc/280409
ER -