HAL CCSD
Intersection Cohomology, Simplicial Blow-Up and Rational Homotopy
Saralegi-Aranguren, Martintxo
Chataur, David
Tanre, Daniel
Laboratoire de Mathématiques de Lens (LML) ; Université d'Artois (UA)
Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 (LAMFA) ; Université de Picardie Jules Verne (UPJV)-Centre National de la Recherche Scientifique (CNRS)
Laboratoire Paul Painlevé (LPP) ; Université de Lille-Centre National de la Recherche Scientifique (CNRS)
International audience
Memoirs of the American Mathematical Society
American Mathematical Society
hal-03620352
https://u-picardie.hal.science/hal-03620352
https://u-picardie.hal.science/hal-03620352
Memoirs of the American Mathematical Society, 2018, 254 (1214), pp.1+. ⟨10.1090/memo/1214⟩
ARXIV: 1205.7057
info:eu-repo/semantics/altIdentifier/arxiv/1205.7057
DOI: 10.1090/memo/1214
info:eu-repo/semantics/altIdentifier/doi/10.1090/memo/1214
en
[MATH]Mathematics [math]
info:eu-repo/semantics/article
Journal articles
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.
2018