# Confusion about a proof from Goresky and MacPherson's “Intersection Homology II”

### 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?

• Depending on the spaces and the definition of the cone, $cone(A\ast B)$ is homeomorphic to $cone(A)\times cone(B)$, as explained in Ronnie Brown's answer here: mathoverflow.net/questions/91790/… – Chris Gerig Apr 18 '18 at 1:25

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.