309 research outputs found
Basic cohomology of canonical holomorphic foliations on complex moment-angle manifolds
We describe the basic cohomology ring of the canonical holomorphic foliation
on a moment-angle manifold, LVMB-manifold or any complex manifold with a
maximal holomorphic torus action. Namely, we show that the basic cohomology has
a description similar to the cohomology ring of a complete simplicial toric
variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and
Zaffran, who previously computed the basic Betti numbers for the canonical
holomorphic foliation in the case of a shellable fan. Our proof uses an
Eilenberg-Moore spectral sequence argument; the key ingredient is the formality
of the Cartan model for the torus action on a moment-angle manifold. We develop
the concept of transverse equivalence as an important tool for studying smooth
and holomorphic foliated manifolds. For an arbitrary complex manifold with a
maximal torus action, we show that it is transverse equivalent to a
moment-angle manifold and therefore has the same basic cohomology.Comment: 16 pages, final published versio
Topological moduli space for germs of foliations II : universal deformations
This work deals with the topological classification of singular foliation germs on (C2,0). Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the projective holonomy representations and we prove the existence of a topological universal deformation through which every equisingular deformation uniquely factorizes up to topological conjugacy. This is done by representing the functor of topological classes of equisingular deformations of a fixed foliation. We also describe the functorial dependence of this representation with respect to the foliation
Homogeneous compact geometries
We classify compact homogeneous geometries of irreducible spherical type and
rank at least 2 which admit a transitive action of a compact connected group,
up to equivariant 2-coverings. We apply our classification to polar actions on
compact symmetric spaces.Comment: To appear in: Transformation Group
Cohomology of lie groups made discrete
We give a survey of the work of Milnor, Friedlander, Mislin, Suslin, and other authors on the Friedlander-Milnor conjecture on the homology of Lie groups made discrete and its relation to the algebraic K-theory of fields
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