Mathematics – Algebraic Topology
Scientific paper
2002-10-02
Extr. Math. 18, No.2, 223-234 (2003)
Mathematics
Algebraic Topology
9 pages article, Latex, amsart-style format. MSC Keywords: Intersection homology, Stratified pseudomanifolds. The format of th
Scientific paper
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.
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