On Normal Stratified Pseudomanifolds

TL;DR

This paper improves the theory of stratified pseudomanifolds by constructing a functorial normalization that preserves intersection homology and satisfies stronger local triviality conditions, extending previous results to broader settings.

Contribution

It proves the existence of a stronger, locally trivial normalization for stratified pseudomanifolds, extending Borel's results to more general cases and providing an explicit, functorial construction.

Findings

Constructed a normalization satisfying stronger local triviality conditions.

Extended Borel's normalization result to broader classes of stratified pseudomanifolds.

Provided an explicit, functorial normalization construction.

Abstract

A stratified pseudomanifold is normal if its links are connected. A normalization of a stratified pseudomanifold $X$ is a normal stratified pseudomanifold $Y$ together with a finite-to-one projection $n:Y\to X$ satisfying a local condition related to the fibers. The map n preserves the intersection homology. Following Borel any pl-stratified pseudomanifod has a normalization in the above sense. In this parper: 1.- We prove that the map $n$ can be required to satisfy a stronger condition: it is a locally trivial stratified morphism preserving the conical structure transverse to the strata. 2.- We extend Borel's result for any topological stratified pseudomanifold and for a family of perversities which is larger than the usual one. 3.- We make an explicit construction of such a normalization. We give a detailed description of the normalizer's stratification. 4.- We prove that our construction is functorial, thus unique up to isomorphisms. With little adjust our procedure holds also in the $C^{\infty}$ category.