Talk:Thom–Mather stratified space
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Alternative definitions of stratified spaces
[edit]mu-stratification and complex analytic stratification as in Kashiwara-Schapira/Dimca
[edit]Sorry, I can't provide a draft, but this would be very nice to have on this page. Maybe it's even better to start a new article, like "analytically stratified space". --Konrad (talk) 14:51, 21 January 2010 (UTC)
Pflaum's approach for stratified pseudomanifolds
[edit]While I find this definition really adjusted to the MacPherson's original viewpoint, I think that in the last years there have raised some alternative approaches to stratified spaces. Between all them, I really like M. Pflaum's way which separates the "stratified" and "conical" features. So, as a comment to this article I would say that it is more a definition of stratified pseudomanifolds (and not so clear, therefore). Here there is a more accurate definition, at least in my opinion.
A stratified space, according to Pflaum, is just a Hausdorff second countable and locally compact topological space X which can be decomposed in disjoint smooth pieces. This means that we give a partition S of X whose elements are locally closed smooth manifolds, we call them strata. The strata must behave well with respect to the incidence and intersetcion problems, so S is asked to be a poset. Two strata R,T in S satisfy R<=T if the closure of R intersects T. We can be even more accurate by saying that the former is the definition of a decomposed space. This is a matter to discussion, since the difference between decompositions and stratifications can be fixed.
In the MacPherson's stratified pseudomanifods; the strata are the diferences Xi+i-Xi between sets in the filtration. There is also a local conical condition; there must be an almost smooth atlas where locally each little open set looks like the product of two factors Rnx c(L); an [|euclidean] factor and the topological cone of a space L. Classically, here is the point where the definitions turns to be obscure, since L is asked to be a stratified pseudomanifold. The logical problem is is avoided by an inductive trick which makes different the objects L and X.
The changes of charts or cocycles have no conditions in the MacPherson's original context. Pflaum asks them to be smooth, while in the Thom-Mather context they must preserve the above decomposition, they have to be smooth in the euclidean factor and preserve the conical radium.
To summarize; stratified spaces are nice topological spaces that can be decomposed in smooth well-behavioured pieces. MacPherson's stratified pseudomanifolds are stratified spaces which locally behave almost like manifolds, that is, almost euclideally and a bit conically. Thom-Mather stratified pseudomanifols also state conditions on the cocycles in order to preserve the conical radium.
Pflaum, M. Analytic and geometric study of stratified spaces. Lecture Notes in Mathematics. vol. 1768. Springer-Verlag. 2001.
- We now have stratified space and so materials here should be merged into that article. I have already put pseudomanifold stuff into that article. —- Taku (talk) 07:56, 13 August 2022 (UTC)