I am currently working on the PL-collapsibility of a particular finite simplicial complex.
Here is my definition of "PL collapsible": I say that there is an elementary collapse from $X$ to $Y$ if there is a ball $B^n\subset X$ and a ball $B^{n-1}\subset\partial B^n$ such that $X = Y\cup B^n$ and $B^{n-1} = Y\cap B^n$. I say that $X$ collapses to $Y$ if there is a sequence of elementary collapses from $X$ to $Y$.
Now here is my question: I have a simplicial complex $X$ that I know to be collapsible (ie, it collapses to a single point), and I consider the simplicial complex $Y$ obtained by deletion of a vertex $x$ whose link in $X$ is a (closed) ball. My question is does this imply that $Y$ is collapsible?
I know that the condition on the link of $x$ implies that $X$ collapses to $Y$. Intuitively, it might happen that this particular way of collapsing $X$ to $Y$ does not necessarily correspond to the start of the known collapse from $X$ to a point, and there might exist "bad ways" to start collapses, that cannot be completed into a full fledged collapse of $X$. But trying out examples, it seems that there is actually no bad way of collapsing, and that would be really helpful for me if there weren't any. Do you have any idea on how to show that such bad ways do not exist, or a counter example showing that they might happen?