# Annulus theorem for pseudomanifolds

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 $$ is the face of some $$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...