Is this simplicial complex homeomorphic to a disk?

Question

Take two $n$-simplices $\sigma_1$ and $\sigma_2$ and join them along a $(n-2)$-dimensional face of each. Is this object homeomorphic to the closed disk $D^n$? How to prove this?

Answer 1

If $n=1$, then you have the disjoint union of two 1-simplices, so maybe you want to assume that $n \geq 2$. Well, if $n=2$, then you're talking about a space $X$ which consists of two triangles attached at a single vertex. That is not homeomorphic to $D^2$: you can remove a single point from $X$ and obtain a non-path connected space, but there are no such points in $D^2$. In general, if you look at points in the $(n-2)$-dimensional face along which you are doing the gluing, their neighborhoods won't look like open sets in $D^n$.