Given a pseudomanifold $X$, an isolated conical singularity $s \in X$ is one whose only possible neighborhoods are of the form $N_s \simeq cL$, where $cL$ denotes the open cone on a compact space $L$. Namely, $s$ is the cone point of some $cL$. Would it be reasonable to assume that one could show that the local homology $H_\bullet(X, X \setminus\{s\}) \ncong H_\bullet(X, X \setminus\{p\})$ for all $p \in N_s$ (of course $p \neq s$)? I haven't come across a proof in the literature, but my intuition is telling me that this is in fact true.
For example, assuming $X$ is Hausdorff, I can find a neighborhood $U$ of $p$ such that $s \not\in U$. Then, clearly $H_\bullet(X, X \setminus\{p\}) \cong \Bbb{Z}$ ($p$ is in the "nice" part of the pseudomanifold $X$). On the other hand, we have $H_\bullet(X, X \setminus\{s\}) \cong H_\bullet(cL, cL \setminus\{s\}) \cong \widetilde{H}_\bullet(cL \cup \bar{c}(L \times (0,1)))$. This is where I'm stuck. I tried interpreting the union $cL \cup \bar{c}(L \times (0,1))$ as a homotopy pushout (as a mapping cone) to take a categorical approach, but to no avail.
As long as $L$ isn't trivial (i.e. Euclidean), I get the feeling that in general there will be more generators for the local homology in a neighborhood of the singularity than at any other point. I need guidance in the formalism.
You should look at the long exact sequence of the pair $(cL,sL\setminus \{s\}$ to compute the relative homology. Since $cL$ is contractible the long exact sequence breaks down and relates the homology of $H_*(cL,cL\setminus \{s\})$ to that of $H_{*-1}(cL\setminus \{s\})\cong H_{*-1}(L)$ for $*>2$. Furthermore $H_1(cL,cL\setminus \{s\})=\mathbb{Z}^{n-1}$, where $H_0(L)\cong \mathbb{Z}^n$. The moral is that you will see all the homology of $L$ in the local homology group.