Intersection Cohomology. Simplicial Blow-up and Rational Homotopy

TL;DR

This paper develops a simplicial approach to intersection cohomology of pseudomanifolds, introduces perverse local systems, and extends rational homotopy theory to include intersection cohomology, creating new topological invariants.

Contribution

It introduces a simplicial blow-up method for intersection cohomology, defines perverse local systems, and extends Sullivan's rational homotopy theory to the intersection setting.

Findings

Constructed a cochain complex isomorphic to intersection cohomology.

Established a functor to perverse cdga's and proved the existence of minimal models.

Proved topological invariance of the minimal model for certain pseudomanifolds.

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 unicity 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.