Martintxo Saralegi-Aranguren

Intersection Cohomology. Simplicial Blow-up and Rational Homotopy

Let X be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. nWe do it simplicially in the setting of a filtered version of face sets, also called simplicial sets without degeneracies, in the sense of C.P. Rourke and B.J. Sanderson. We define perverse local systems over filtered face sets and intersection cohomology with coefficients in a perverse local system. In particular, as announced above when X is a pseudomanifold, we get a perverse local system of cochains quasi-isomorphic to the intersection cochains of Goresky and MacPherson, over a field. We show also that these two complexes of cochains are quasi-isomorphic to a filtered version of Sullivans differential forms over the field Q. In a second step, we use these forms to extend Sullivans presentation of rational homotopy type to intersection cohomology. nFor that, we construct a functor from the category of filtered face sets to a category of perverse commutative differential graded Q-algebras (cdgas) due to Hovey. We establish also the existence and unicity of a positively graded, minimal model of some perverse cdgas, including the perverse forms over a filtered face set and their intersection cohomology. Finally, we prove the topological invariance of the minimal model of a PL-pseudomanifold whose regular part is connected, and this theory creates new topological invariants. This point of view brings a definition of formality in the intersection setting and examples are given. In particular, we show that any nodal hypersurface in CP(4), is intersection-formal.

Tautness for riemannian foliations on non-compact manifolds

For a riemannian foliation

Poincaré duality with cap products in intersection homology

{mathcal{F}}

Lefschetz duality for intersection (co)homology

on a closed manifold M, it is known that

Equivariant intersection cohomology of the circle actions

{mathcal{F}}

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

is taut (i.e. the leaves are minimal submanifolds) if and only if the (tautness) class defined by the mean curvature form