• Corpus ID: 237373879

Simlpicial intersection homology revisited

@inproceedings{Chataur2021SimlpicialIH,
  title={Simlpicial intersection homology revisited},
  author={David Chataur and Martintxo E. Saralegi-Aranguren and Daniel Tanr'e},
  year={2021}
}

• D. Chataur, M. Saralegi-Aranguren, Daniel Tanr'e
• Published 1 September 2021
• Mathematics

Intersection homology is defined for simplicial, singular and PL chains. In the case of a filtered simplicial complex, it is well known that the three versions are isomorphic. This isomorphism is established by using the PL case as an intermediate between the singular and the simplicial situations. Here, we give a proof similar to the classical proof for ordinary simplicial complexes. We also study the intersection blown-up cohomology that we have previously introduced. In the case of a pseudomanifold, this cohomology owns a Poincaré isomorphism with the intersection homology, for any coefficient ring, thanks to a cap product with a fundamental class. We prove that the simplicial and the singular blown-up cohomologies of a filtered simplicial complex are isomorphic. From this result, we can now compute the blown-up intersection cohomology of a pseudomanifold from a triangulation. Finally, we introduce a blown-up intersection cohomology for PL-spaces and prove that it is isomorphic to the singular ones. We also show that the cup product in perversity 0 of a CS-set coincides with the cup product of the singular cohomology of the underlying topological space. The cohomology of a triangulated manifold can be computed in many different ways : using the simplices of the triangulation, using the simplicial set of singular simplices, using sheaf theory, using homotopy classes into Eilenberg-MacLane spaces. This paper and our previous works on blown-up cochains [4, 7, 8] show that we have the same picture for intersection cohomology. Furthermore, we show the existence of an isomorphism between the singular and simplicial definitions of intersection homology, without involving the PL structure and without limitation to CS sets. Let us recall some historical background of intersection homology as it was introduced by Goresky and MacPherson. In their first paper on intersection homology, they introduce it for a pseudomanifold X together with a fixed PL-structure and a parameter p, now called GoreskyMacPherson perversity. They define the complex of p-intersection, IC p ̊pXq, as a subcomplex of the complex of PL-chains and the p-intersection homology as the homology of IC p ̊pXq. Let R be a commutative ring of coefficients. We could also start with a fixed triangulation on a stratified Euclidean simplicial complex, K, and define a complex of p-intersection, C p ̊pKq, as a subcomplex of the simplicial chains. This is, for instance, the concrete case in [16], where MacPherson and Vilonen give a construction of the category of perverse constructible sheaves on a Thom-Mather stratified space with a fixed triangulation. In an appendix to this paper, Goresky and MacPherson prove that the two intersection homologies are isomorphic; i.e., the simplicial one on K and the previous PL one on the PL space X associated to K. They do that with a nice construction of a left inverse to the inclusion C p ̊pKq Ñ IC p ̊pXq, see the book of Friedman ([9, Section 3.3]) for a detailed version of this argumentation. A third step is to consider the topological realization, |K|, of K and define a complex of p-intersection, C p ̊p|K|q, as a subcomplex of the singular chains on |K|. This situation has been studied by King [15] who proves the existence of an isomorphism between the two homologies, in the case of CS sets. This situation is developed in a thorough way in [9, Section 5.4]. Let us emphasize that the proof is not a direct comparison between the simplicial and the singular Date: September 2, 2021. 2010 Mathematics Subject Classification. 55N33, 55N35.