Face map not well-defined on oriented simplices

Question

Currently I am facing this problem with the face map $d_i$ (delete the $i$th vertex). It is not well-defined on oriented simplices.

For instance, $d_0[v_0,v_1,v_2]=[v_1,v_2]$, but $d_0[v_2,v_0,v_1]=[v_0,v_1]$.

Is there a known way to ensure that this does not happen?

I thought of the following: we order the vertices and impose the rule of always writing simplices in ascending order of vertices, with a negative sign if necessary.

For instance $d_0(-[v_0,v_1,v_2])=-[v_1,v_2]$.

In essence, I wish to find a working definition such that the $i$-th face of any simplex $\sigma$, $d_i\sigma$, is well-defined.

Does the method outlined above work? Is there a name for such construction? Thanks.

Answer 1

The usual approach here is to define a "simplex" to come with a specific ordering of the vertices. So, you would simply say that $[v_0,v_1,v_2]$ and $[v_2,v_0,v_1]$ are different simplices, and there is no problem with $d_0$ taking different values on them.

If you are talking about the simplices of a simplicial complex, then it would furthermore be usual to fix a single total order of the vertex set, and allow only simplices whose vertices are in that order. So, only $[v_0,v_1,v_2]$ would be considered a simplex in your complex at all, and $[v_2,v_0,v_1]$ would be a meaningless symbol. This is basically equivalent to your proposed solution (the only difference being that in your version you also consider something like $-[v_0,v_1,v_2]$ to be a "simplex", rather than just being the formal negative of a simplex).

Answer 2

My answer is based on what you commented of doing homology.

Let $K$ be a simplicial complex, $\sigma$ a $n-$simplex, with vertices $v_0, ..., v_n$. Let $\pi$ be a permutation of $\{1, ... , n\}$. We say that the orderings $(v_0, ..., v_n)$ and $(v_{\pi(0)}, ..., v_{\pi(n)})$ are equivalent if $\pi$ is an even permutation.

So, we define an oriented simplex $[v_0, ..., v_n]$ to be the class of the ordering $(v_0, ..., v_n)$.

Then one constructs $C_n(K)$ to be the group of functions from the set of oriented $n-$simplices of $K$ to $\mathbb{Z} $ such that $c(\sigma)=-c(\sigma')$ for $\sigma, \sigma'$ different orientations of the same simplex and $c(\sigma)\neq 0$ only for finite simplices $\sigma$. This is equivalent to say, $C_n(K)$ to be the free abelian group generated by oriented $n-$simplices of $K.$

The vertices are labelled so $d_i$ removing $v_i$ means exactly that (not removing the vertex in the $i-$th place).

I write $\partial_n:C_n(K)\rightarrow C_{n-1}(K)$, for an ordering $(v_0, ..., v_n)$ of $\sigma$, $$\partial_n(v_0, ..., v_n)=\sum_{i=1}^{n} (-1)^n[v_0, ...., \hat{v_i}, ..., v_n],$$ where the "hat" means remove that vertex, that is what you wanted to call $d_i$.