The notion of cs-stratification of a topological space is apparently due to Siebenmann, see also the paper by N. Habegger and L. Saper in the paper "Intersection cohomology of cs-spaces and Zeeman's filtration", Invent.Math. (1991).

First I remind its definition and then formulate my question.

Definition. Let $X$ be a topological space. A filtration $\mathcal{X}$ of $X$ by closed subsets $$X=X_n \supset X_{n-1}\supset\dots \supset X_0\supset X_{-1}=\emptyset$$ is called cs-stratification of dimension $n$ if it satisfies the following properties:

(i) The stratum $S_{n-k}:=X_{n-k}\backslash X_{n-k-1}$ is either empty or a $(n-k)$-dimensional topological manifold.

(ii) $S_n=X_n\backslash X_{n-1}$ is non-empty.

(iii) (Local Normal Triviality) For any $x\in S_{n-k}$ there exists a neighborhood $U$ of $x$ in $X$ and a homeomorphism \begin{eqnarray}\label{E:1} U=B\times c(L),\,\,\, (\ast) \end{eqnarray} where $B$ is an open $(n-k)$-dimensional ball, and $c(L)$ is the open cone \begin{eqnarray*} c(L)=L\times [0,1)/(x,0)\sim(x',0),\\ (c(\emptyset)\mbox{ is a point}) \end{eqnarray*} on a compact topological space $L$ (called the link). $L$ is assumed to have a filtration $\mathcal{L}$ $$L=L_{k-1}\supset L_{k-2}\supset \dots\supset L_0\supset L_{-1}=\emptyset$$ for which the homeomorphism ($\ast$) satisfies $$U\cap X_{n-j}=B\times c(L_{k-j-1}).$$

As far as I understand it follows automatically that any subset $L_k\subset L$ is closed. But my question is as follows.

Question. It is not explicitly mentioned in the definition that the filtration $\mathcal{L}$ of $L$ is a cs-stratification (by induction on the dimension). Was this assumption omitted by purpose or it should be added? Does it follow automatically from the other assumptions?


You should look at Greg Friedman's book: http://faculty.tcu.edu/gfriedman/IHbook.pdf CS-sets are discussed in section 2.3. In fact for many purposes it is rather interesting to suppose that the links $L$ are just compact filtered topological spaces and not CS-sets. Deligne's axioms for intersection homology sheaves are easy to verify in this setting where we just suppose that links are compact.

The definition does not imply $L$ is itself a CS-set.

Such a recursive assumption is useful when you want to study pseudomanifolds and Poincaré duality of pseudomanifolds in intersection homology.

  • 1
    $\begingroup$ In Siebenmann's "Deformation of Homeomorphisms on Stratified Sets," where he introduces CS sets, he explicitly notes on page 128 that he does not make this assumption. You can still prove a surprising amount about these spaces without such an assumption! (And thanks, David, for referencing my book!) $\endgroup$ – Greg Friedman Jul 30 '16 at 7:26

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