Let $K$ be a simplicial complex. An elementary simplicial collapse is a formal operation on $K$ involving removing a free face (i.e. a simplex with a unique cofacet) and its unique cofacet. If $L \subset K$ is a subcomplex, a simplicial collapse $K \searrow L$ is the composition of a finite sequence elementary simplicial collapses such that every simplex in $K\setminus L$ is removed by some elementary collapse in the sequence. I am interested in understanding the failure of the 2-out-of-3 property for of simplicial collapses for finite simplicial complexes.
- The composition of simplicial collapses is clearly a simplicial collapse.
- If $M \subset L \subset K$ are finite simplicial complexes such that $K \searrow M$ and $K \searrow L$ then it is not necessary that $L \searrow M$. There is an explicit counter-example where $M$ is a triangulation of the 3-ball, $L$ is a triangulation of the dunce hat, and $K$ is a point.
I suspect that the third counterpart to these two properties is also false, but I have not been able to come up with a counter-example.
Question: I am looking for an explicit counter-example to the following claim: if $M \subset L \subset K$ are finite simplicial complexes such that $K$ and $L$ simplicially collapse to $M$ then $K$ simplicially collapses to $L$. That is, I want to find complexes where no collapse of $K$ onto $M$ factors through a collapse of $L$ onto $M$.