Proof review of a theorem

Question

I have the following proof of a little lemma and I want to know if it is clear in its structure and if it actually provides the claimed results.

$\mathcal K$ is an abstract simplicial complex.

Let $\tau, \sigma \in \mathcal K$ such that the following two conditions are satisfied:

1) $\tau \subset \sigma$, in particular dim $\tau<$ dim $\sigma$.

2) $\sigma$ is a maximal face of $\mathcal K$ and no other maximal face of $\mathcal K$ contains $\tau$.

Then $\tau$ is called a free face. A simplicial collapse of $\mathcal K$ is the removal of all simplices $\gamma$ such that $\tau \subseteq \gamma \subseteq \sigma$, where $\tau$ is a free face. If additionally we have dim $\tau = $ dim $\sigma-1$, then this is called an elementary collapse. An elementary collapse at $\tau$ is a dim$(\tau)-$collapse.

A simplicial complex $\mathcal K$ collapses to a simplicial complex $\mathcal L$ if there is a sequence of simplicial complexes $\mathcal K = \mathcal K_0 \searrow \mathcal K_1 \searrow \cdots \searrow \mathcal K_q = \mathcal L$ such that $\mathcal K_{i+1}$ is obtained from $\mathcal K_i$ by a collapse at $\tau_i$. In this case, we call $C = (\tau_o, \tau_1,\cdots,\tau_{q-1})$ a $(\mathcal K,\mathcal L)$ collapsing sequence and write $\mathcal K \searrow^C \mathcal L$. A simplicial complex is called collapsible if it collapses to a single vertex.

A collapsing sequence $C$ is monotone if $i < j$ imply $\text{dim}(\sigma_i) \geq \text{dim}(\sigma_j)$. In this case, if $\mathcal K \searrow \mathcal L$, we say that $\mathcal K$ monotonically collapses to $\mathcal L$.

Now I want to prove that: If $\mathcal K$ collapses to $\mathcal L$, then $\mathcal K$ monotonically collapses to $\mathcal L$.

This is my proof: Consider a collapse sequence $C=(\tau_0, \tau_i,\cdots,\tau_n)$. Starting form $C$ we can construct explicitly a monotone collapse sequence $C'$ by specifying at each step how to choose a new simplex $\tau'$. Denote by $d$ the dimension of $\mathcal K$. We describe the construction with respect to dimension $d$, and as $d$ decreases we cover the entire sequence. Let $\tau \in C$ and $\sigma$ the unique maximal face containing it. Only the following cases are possible:

1) $\text{dim}(\tau) = d-1, \text{dim}(\sigma)=d \Rightarrow \tau'=\tau$;

2) $\text{dim}(\tau) < d-1, \text{dim}(\sigma)=d \Rightarrow \tau'=\gamma$ with $\gamma$ such that $\gamma \supset \sigma$, $\text{dim}(\gamma)=d-1$. $\gamma$ is free since $\gamma \supset \tau$ and $\tau$ is free. For a simplex there exists a monotonically collapse sequence. Collapse all simplicies $\delta \supset \sigma$ of dimension $\text{dim}(\delta)$ at the beginning of step $d = \text{dim}(\delta)$, following the order in which we have found at this step.

3) $\text{dim}(\tau) < d$. Collapse $\tau$ at the beginning of step $d = \text{dim}(\tau)$, following the order in which we have found at this step.

At step $d$, all the simplicies $\tau'$ have dimension $d$. In addition, in 2) we have removed a maximal face of dimension $d$ and in 3) such a face is not present at all; at step $d$ only of the faces of dimension $d$ can prevent us to continue with collapses, thus we can actually swap 3) and remove only $\gamma$ in 2). At the beginning of every successive step we first collapse the simplicies from point 2) and 3). Finally, if $\text{dim}(\tau)=d'<d$ at step $d$, then at step $d'$, $\tau$, satisfy condition 1) of above.

Please comment for suggestions or improvements in the explanation

Answer 1

As written, your proof seems unclear to me. Perhaps there is some kind of induction going on, but I am not clear on that point, because I do not see a clearly formulated induction proof, in particular I do not see a clearly stated induction hypothesis.

If you were doing an induction, I would have expected that you are using it to construct a collapsing sequence $C' = (\tau'_0,\tau'_1,...,\tau'_n)$ from $\mathcal K$ to $\mathcal L$, and I would have expected an induction hypothesis describing the properties of an initial subsequence $(\tau'_0,\tau'_1,...,\tau'_{j-1})$ and of the corresponding sequence $\mathcal K = \mathcal K'_0 \searrow \mathcal K'_1 \searrow \cdots \searrow \mathcal K'_j$; presumably "monotonicity" would be part of the induction hypothesis.

In carrying out the induction step, you would then be required to prove the existence of a free face $\tau'_j$ of $\mathcal K'_j$ such that the maximal simplex $\sigma'_j$ of $\mathcal K'_j$ that contains $\tau'_j$ has dimension equal to the dimension of $\mathcal K'_j$.

In particular, at the point where you say "Let $\tau \in C$ and $\sigma$ the unique maximal face containing $\tau$", I cannot tell at all where in the induction you are and what role $\tau$ will play in the induction step. What if $\tau$ has already been removed by the time you get to $\mathcal K'_i$? What if $\tau$ has not been removed yet and is not a free face of $\mathcal K'_i$?

My question in the comments was aimed at these issues. If, as you say in your comment, $\tau=\tau_i$ for some $i$, and if $\sigma_i$ is the unique maximal face of $\mathcal K_i$ that contains $\tau_i$, and if $\tau_i = \tau'_j$ is also the free face of some $\mathcal K'_j$ that you intend to collapse in the induction step, then it is not at all clear that the maximal simplex of $\mathcal K'_j$ that contains $\tau_i$ is equal to the maximal simplex of $\mathcal K_i$ that contains $\tau_i$, nor that it even has the same dimension.