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The BIC of a singular foliation defined by an abelian group of isometries

100%

Saralegi-Aranguren M., Wolak R.

Annales Polonici Mathematici

2006

tom 89

nr 3

203-246

We study the cohomology properties of the singular foliation ℱ determined by an action Φ: G × M → M where the abelian Lie group G preserves a riemannian metric on the compact manifold M. More precisely, we prove that the basic intersection cohomology $ℍ*_{p̅}(M/ℱ)$ is finite-dimensional and satisfies the Poincaré duality. This duality includes two well known situations: ∙ Poincaré duality for basic cohomology (the action Φ is almost free). ∙ Poincaré duality for intersection cohomology (the group G is compact and connected).

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

81%

Royo Prieto J., Saralegi-Aranguren M., Wolak R.

Bulletin of the Polish Academy of Sciences. Mathematics

2005

tom 53

nr 4

429-440

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.

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Bulletin of the Polish Academy of Sciences. Mathematics

2 Saralegi-Aranguren M.

2 Wolak R.

1 Royo Prieto J.

1 2006

1 2005

Wyniki wyszukiwania

The BIC of a singular foliation defined by an abelian group of isometries

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