Elementary collapses implies same homotopy type

Question

Let $\Delta$ be a simplicial complex, and suppose that $\sigma \in \Delta$ is a proper face of exactly one maximal simplex $\tau \in \Delta$. A simplicial collapse of $\Delta$ is the removal of all simplices $\gamma$ s.t. $\sigma \subseteq \gamma \subseteq \tau.$. It is well known that a sequence of collapses yields a strong deformation retraction, in particular, a homotopy equivalence. I have an intuitive geometric feeling of how to prove this fact, but I don't know how we can construct this deformation retract map. Can anyone give the idea for the deformation retract map?

Answer 1

Such $\sigma$'s are called free faces. You can think of these collapses as "pushing in" the free face. For example, if your simplicial complex is a triangle $\Delta^2$, then any edge $\sigma$ is a free face as it is contained in the unique $2$-simplex $\tau$. Pushing in $\sigma$ through $\tau$ collapses those two simplices onto the remaining two edges. This is a strong deformation retract as all other faces remain fixed through this process.