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Disclaimer: I don't fully understand what I'm talking about in the question below. I'm still trying to figure out the right question to ask. Quotations and question marks in brackets mean that I'm not sure what the right notion should be, or if what I'm saying makes reasonable sense. At any rate, I'm interested in a certain space formed as follows...

So, let $X$ be a (stratified?) manifold of dimension $n$, and define

  • $X_0 := \emptyset$
  • For $k=1,\ldots,n$, let $X_k$ be the set of points $x \in B_k := X\setminus \left(\cup_{j=0}^{k-1}X_j\right)$ which have a neighborhood $\mathcal N_x$ such that $B_k \cap \mathcal N_x$ "looks like" a hyperplane of codimension $k$.

Suppose that the decomposition holds: $X=\cup_{k=1}^nX_k$ (otherwise simply consider the space $X':=\cup_{k=1}^n X_k$ in all what follows ?).

Question

  • What topological / geometric conclusions can be drawn about the above scenario ? In particular
  • How do the "Betti numbers" of $X$ relate to those of the "pieces" $X_k$ ?
  • How is the "(co)homology" of $X$ related to that of the pieces $X_k$ ?
  • What area of topology / geometry is concerned with such questions ?
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  • $\begingroup$ In what situation do you encounter this? $\endgroup$ – Thomas Rot Jul 9 at 16:17
  • $\begingroup$ In trying to understand the complexity (e.g number of linear pieces, etc.), of a random multivariate piecewise linear funciton. $\endgroup$ – dohmatob Jul 9 at 16:20
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    $\begingroup$ I would advise looking for literature on pseudomanifolds and intersection homology. The Wikipedia article has a lot of references. (Though this depends on exactly how you "stratified manifold" looks like. This may not necessarily be adapted.) $\endgroup$ – Najib Idrissi Jul 9 at 16:46
  • $\begingroup$ Thanks for the references. Will dig them up. $\endgroup$ – dohmatob Jul 9 at 16:54
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    $\begingroup$ Another reference: "The topological classification of stratified spaces" by Weinberger. $\endgroup$ – Paul Siegel Jul 9 at 16:59

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