Annulus theorem for pseudomanifolds

Question

Lets say I take an arbitrary closed and smooth $d$-manifolds $\mathcal{M}$. Now, it is a well-known fact that whenever I take two (sufficiently nice embedded) closed $d$-balls $B_{1}$ and $B_{2}$ in $\mathcal{M}$, then the manifold obtained by cutting out the interior of $B_{1}$ is homeomorphic to the manifold obtained by cutting out the interior of $B_{2}$ from $\mathcal{M}$. In other words, for a closed manifold, it does not depend on where I cut out a ball (up to homeomorphism). This is basically a consequence of the famous and highly non-trivial annulus theorem. For dimension 2 and 3, it follows from the famous triangulation theorems by Radó (1924) and Moise-Bing (1952,1959), but for higher dimension it was only proven in 1969 by Kirby ($d>4$) and in 1982 by Quinn (for d=4).

I am wondering if some similar statement is true for pseudomanifolds. By a pseudomanifold, I mean the following:

Let $\Delta$ be a finite abstract $d$-dimensional simplicial complex. Its geometric realization $\vert\Delta\vert$ is $d$-dimensional pseudomanifold (without boundary), if the following conditions are fulfilled:

1. It is "pure", i.e. every simplex $\sigma\in\Delta$ of dimension $<d$ is the face of some $d$-simplex.
2. It is "non-branching", i.e. every $(d-1)$-simplex is face of exactly two $d$-simplices.
3. It is "strongly connected", i.e. for every two $d$-simplices $\sigma,\tau\in\Delta_{d}$, there is a sequence of $d$-simplices $\sigma=\sigma_{1},\sigma_{2},\dots,\sigma_{k}=\tau$ such that $\sigma_{l}\cap\sigma_{l+1}$ is a $(d-1)$-simplex $\forall l$.

Obviously, every PL-manifold is a pseudomanifold, but not vice versa. A famous example is the pinched torus, which is obtained by identifying two distinct points of the $2$-sphere.

So, my question is,

when I remove a closed $3$-ball inside a closed pseudomanifold (for example, by deleting the interior of a single $3$-simplex or any subcomplex PL-homeomorphic to it), does the result depend on where I cut the ball (up to PL-homeomorphism)?

I am mostly interested in the $3$-dimensional case, which is usually a little bit easier than the higher-dimensional cases...

Answer 1

The homeomorphism type of the space left after deleting the interior of the ball can depend on whether the ball intersects the singular locus (necessarily in the boundary of the ball) or not. If your ball intersects the singular locus, then once you remove the ball interior such points need not have neighborhoods homeomorphic to Euclidean half space, which will be the case if you remove the interior of a simplex in the interior of the manifold part of the pseudomanifold. On the other hand, if you restrict the balls to lie entirely disjoint from the singular locus, then your problem occurs entirely within the manifold obtained by removing the singular locus, in which case you can apply the classical results.

As a concrete example, suppose you triangulate the torus and then take your pseudomanifold to be the suspension of that torus. Let your ball be one of the cones on one of the 2-simplices of the torus. When you remove the interior of the ball, the cone point now has a neighborhood homeomorphic to the cone on a torus with an open disk removed. A local homology argument can then be used to show that the cone point does not have a neighborhood homeomorphic to Euclidean half-space.