Confusion about a proof from Goresky and MacPherson's "Intersection Homology II"

Question

Context

My question is about the "proof of claim" on page 84 of Goresky and MacPherson's "Intersection Homology II". For ease of reading, here's the claim:

Claim: Suppose $X$ is a topological pseudomanifold$^*$ and $\mathbf A^\bullet$ is a topologically constructible$^*$ complex of sheaves on $X$. For any $x\in X$ there is a neighborhood basis $U_1\supset U_2\supset U_3\supset\ldots$ such that for each $i$ and $m$, the restriction map $$H^i(U_m; \mathbf{A}^\bullet) \to H^i(U_{m+1}; \mathbf{A}^\bullet)$$ is an isomorphism.

$^*$ See [$\S$1.1 and $\S$1.4, loc. cit.] for the definitions of topological pseudomanifold and topologically constructible.

My Question

The only part of the proof I'm having issues with is the very beginning, where the authors construct the neighborhood basis itself. They start by choosing a distinguished neighborhood $N\cong \Bbb R^i\times \operatorname{cone}^\circ(L)$ of the point $x$. They then consider the join $Y=S^{i-1}*L$ and assign it a stratification. I'm OK so far. However, they then say, "Choose a stratum preserving homeomorphism $\psi\colon \operatorname{cone}^\circ(Y)\to N$ with $\psi(\text{vertex})=x$."

Question: (1) Why are $\operatorname{cone}^\circ(Y)$ and $N$ homeomorphic, and (2) why does there exist a stratum preserving homeomorphism (with $\psi(\text{vertex})=x$) between them?

Answer 1

As noted by Chris Gerig in the comments, letting $cX$ denote the open cone on the compact space $X$ then $(cX)\times (cY)\cong c(X*Y)$, where $X*Y$ is the join. In the case at hand, Goresky and MacPherson are treating $\mathbb{R}^i$ as $cS^{i-1}$. When $X$ and $Y$ are stratified, there is a natural stratification of the join. I discuss this in Section 2.11 of my book - draft currently available here: http://faculty.tcu.edu/gfriedman/IHbook.pdf

This said, the typical stratification of the cone has the vertex as a $0$-dimensional stratum, which can't be the case here thinking of the distinguished neighborhood $N$ as a cone (unless $i=0$). So I would say there isn't quite a stratum preserving homeomorphism here in the way you would expect from the usual definitions. They must be thinking of $cS^{i-1}$ as a single stratum.

All of this said, if you're more interested in the theorem in whose proof this statement appears, you can find a much more thorough treatment of the constructibilty issues in Borel's book "Intersection Cohomology," particularly Section V.3. In particular, the Claim you cite is contained in Borel's Proposition V.3.10 on page 77.

Note that Goresky and MacPherson make constructibility assumptions, but Borel shows that these assumptions all follow from the definition of the Deligne sheaf and so don't need to be assumed.