Top-Dimensional Group of the Basic Intersection Cohomology
for Singular Riemannian Foliations
Bulletin of the Polish Academy of Sciences. Mathematics, Tome 53 (2005) no. 4, pp. 429-440
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
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 $\mathbb R$.
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.
Keywords:
known regular riemannian foliation compact manifold properties its basic cohomology non vanishing top dimensional group poincar duality tautness foliation closely related consider singular riemannian foliations there little relation between these properties present example singular isometric flow which top dimensional basic cohomology group non trivial basic cohomology does satisfy poincar duality however recover poincar duality basic intersection cohomology accidental top dimensional basic intersection cohomology groups example isomorphic either mathbb prove holds singular riemannian foliation compact connected manifold corollary tautness regular stratum singular riemannian foliation detected basic intersection cohomology
Affiliations des auteurs :
José Ignacio Royo Prieto 1 ; Martintxo Saralegi-Aranguren 2 ; Robert Wolak 3
@article{10_4064_ba53_4_8,
author = {Jos\'e Ignacio Royo Prieto and Martintxo Saralegi-Aranguren and Robert Wolak},
title = {Top-Dimensional {Group} of the {Basic} {Intersection} {Cohomology
for} {Singular} {Riemannian} {Foliations}},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
pages = {429--440},
publisher = {mathdoc},
volume = {53},
number = {4},
year = {2005},
doi = {10.4064/ba53-4-8},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/ba53-4-8/}
}
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 SP - 429 EP - 440 VL - 53 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4064/ba53-4-8/ DO - 10.4064/ba53-4-8 LA - en ID - 10_4064_ba53_4_8 ER -
%0 Journal Article %A José Ignacio Royo Prieto %A Martintxo Saralegi-Aranguren %A Robert Wolak %T Top-Dimensional Group of the Basic Intersection Cohomology for Singular Riemannian Foliations %J Bulletin of the Polish Academy of Sciences. Mathematics %D 2005 %P 429-440 %V 53 %N 4 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4064/ba53-4-8/ %R 10.4064/ba53-4-8 %G en %F 10_4064_ba53_4_8
José Ignacio Royo Prieto; Martintxo Saralegi-Aranguren; Robert Wolak. Top-Dimensional Group of the Basic Intersection Cohomology for Singular Riemannian Foliations. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 53 (2005) no. 4, pp. 429-440. doi: 10.4064/ba53-4-8
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