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Intersection Cohomology, Simplicial Blow-Up and Rational Homotopy

Abstract : 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. We 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 Sullivan's differential forms over the field Q. In a second step, we use these forms to extend Sullivan's presentation of rational homotopy type to intersection cohomology. For that, we construct a functor from the category of filtered face sets to a category of perverse commutative differential graded Q-algebras (CDGA's) due to Hovey. We establish also the existence and uniqueness of a positively graded, minimal model of some perverse CDGA's, 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.
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Soumis le : vendredi 25 mars 2022 - 17:57:58
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Martintxo Saralegi-Aranguren, David Chataur, Daniel Tanre. Intersection Cohomology, Simplicial Blow-Up and Rational Homotopy. Memoirs of the American Mathematical Society, American Mathematical Society, 2018, 254 (1214), pp.1+. ⟨10.1090/memo/1214⟩. ⟨hal-03620352⟩



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