Stratified simplices and intersection homology

Abstract

Intersection homology is obtained from ordinary homology by imposing conditions on how the embedded simplices meet the strata of a space $X$. In this way, for the middle perversity, properties such as strong Lefschetz are preserved. This paper defines local-global intersection homology groups, that record global information about the singularities of $X$. They differ from intersection homology in that stratified rather than ordinary simplices are used. An example of such is $σ_j\times Cσ_i$, where $σ_i$ and $σ_j$ are ordinary simplices, and $C$ is the coning operator. The paper concludes with a sketch of the relationship between local-global homology and the geometry of convex polytopes. This paper is a more formal exposition of part of the author's `Local-global intersection homology', alg-geom/9709011.