# Proof review of a theorem

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 $$\text{dim}(\tau)=d' at step $$d$$, then at step $$d'$$, $$\tau$$, satisfy condition 1) of above.

Please comment for suggestions or improvements in the explanation

• When you write "Let $\tau \in C$ and $\sigma$ the unique maximal face containing it", what do you mean? The maximal face in $\mathcal K$ containing $\tau$ need not be unique, although if $\tau=\tau_i$ then the maximal face in $\mathcal K_i$ containing $\tau_i$ is unique. Aug 15, 2019 at 14:55
• With $\tau \in C$ I mean that $\tau=\tau_i$ for some $i$, $\tau$ is an element of the sequence Aug 16, 2019 at 15:32

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.