An elementary collapse (https://en.wikipedia.org/wiki/Collapse_(topology)) is the removal of a pair of simplices $\sigma,\tau$, such that $\dim \tau=\dim\sigma-1$ and $\tau$ is a free face of $\sigma$.
I am looking for a way to systematically collapse a $n$-simplex $\Delta$=$[v_1,v_2,\dots,v_{n+1}]$ to a point $[v_1]$.
For small $n$, I can check it manually. For instance, for $n=2$, an explicit sequence of elementary collapses is:
1) Remove $[v_1,v_2,v_3]$ and $[v_1,v_3]$
2) Remove $[v_2,v_3]$ and $[v_3]$
3) Remove $[v_1,v_2]$ and $[v_2]$
We are then left with the single vertex $[v_1]$.
Intuitively, for general $n$ we can always collapse the $n$-simplex to a point.
However, I am having trouble coming out with a systematical and explicit sequence of elementary collapses for general $n$. What I am looking for is something like my example for $n=2$ above, an explicit sequence of elementary collapses for general $n$.
Thanks for any help.
For every face $\tau\neq\emptyset$ which does not contain $v_1$ match it with $\tau\cup\{v_1\}$. This is an acyclic matching in the discrete Morse-theoretic sense which gives you the order of collapses. This works for any cone with apex $v_1$.
Or if you like it without matching: in each collapsing step choose a maximal face $\tau$ not containing $v_1$ and remove it together with $\sigma=\tau\cup\{v_1\}$.