Introduction to Intersection Homology

• Grant number: 22/10020-8
• Support Opportunities: Scholarships in Brazil - Master
• Effective date (Start): December 01, 2022
• Effective date (End): July 31, 2024
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Nivaldo de Góes Grulha Júnior
• Grantee: Hana Marinho Lucena
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Associated research grant: 19/21181-0 - New frontiers in Singularity Theory, AP.TEM

Abstract

In topology, intersection homology is analogous to singular homology, especially suited to the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in 1974 and developed by them in the following years.The homology groups of a compact, oriented and connected n-dimensional manifold X have a fundamental property called Poincaré duality. Classically, going back, for example, to HenriPoincaré, this duality was understood in terms of intersection theory. An element of is {\displaystyle H_{j}(X)} represented by a j-dimensional cycle. If an i-dimensional and an (n-i)-dimensional cycle are in general position, then its intersection is a finite collection of points. Using the orientation of X, one can assign each of these points a sign; in other words, the intersection produces a 0-dimensional cycle. It can be proved that the homology class of this cycle depends only on the homology classes of the original cycle (with dimensions i and (n-i)).However, when X has singularities, these ideas lose their meaning. In such a case, for example, it is no longer possible to establish a good notion of "general position" for cycles. For this reason, Goresky and MacPherson presented a class of "permissible" cycles, which makes sense using the general position term.Thus, an equivalence relation for allowed cycles was introduced, and the group {\displaystyle IH_{i}(X)} was called "permissible i-dimensional cycles", modulo this equivalence relation, "intersection homology".Furthermore, it was shown that the intersection of an i- and an (n-i)-dimensional permissible cycle gives a zero cycle (common) whose homology class is well defined.In this work, we intend to systematically organize the introductory theory of homology ofintersection.