Understanding the statement "the boundary of the boundary of a simplex is zero"

Question

Suppose that we have a triangle simplex $S=(p_{1}p_{2}p_{3})$, then the boundary of $S$ $\partial S$ is defined by $\partial S=p_{1}p_{2}+p_{2}p_{3}+p_{3}p_{1}$, then $\partial \partial S=p_{1}-p_{2}+p_{2}-p_{3}+p_{3}-p_{1}=0$,but from wikipidea, the plus sign here is not the sum of vectors or sum of sets, it is used to separate each member in the set and the set consists of finite k-simplexes, so my questions are:(1) what is the meaning of $0$ here(2) why the sum is $0$, as to the second question I think $p$ and $-p$ cancel out but I don't know how to understand the above sum $p_{1}-p_{2}+p_{2}-p_{3}+p_{3}-p_{1}=0$ rigorously

Answer 1

When you talk about boundary operator, then you already have to have chains introduced. Given a simplicial complex $\Delta$, and a ring $R$ we define $C_n$ to be the free module over $R$ with (oriented) $n$-simplexes of $\Delta$ as basis. Elements of $C_n$ are called $n$-chains and are formal sums of $n$-simplexes with formal coefficients taken from $R$. When $R$ is a field then this means vector space, when $R=\mathbb{Z}$ then this means free abelian group.

Either way $C_n$ is an abelian group at least. And thus all rules of the abelian groups apply to it.

So what is a chain? We are not dealing with Euclidean vectors here or some set unions. We are dealing with an abstract construction that follows certain rules. So for example if $A,B$ are two $n$-simplexes then we can form their sum $A+B$. And that's it. That's their sum. It is not another $n$-simplex, it is a formal sum, an expression of the form $A+B$. It is w chain. We can also multiply $n$-simplex by say $5$ (given $R=\mathbb{Z}$). Then we get $5A$. That's it. It is not a simplex $5$ times bigger than $A$, no, it is an expression of the form $5A$. Finally since we deal with an abelian group then it also has a zero. Again, you may think of it as some abstract expression, lets use $\Theta$ symbol to represent it (so that it resembles standard zero, but also says clearly that it is some abstract thing).

Those formal expressions are not without rules. Since we deal with abelian groups, then we have $A+\Theta=A$ for any $n$-simplex $A$. We also have $A+B=B+A$. Furthermore $A-B$ is not a reducible expression, unless $B=A$, in which case $A-A=\Theta$, just like in any group.

I think $p$ and $-p$ cancel out but I don't know how to understand the above sum $p_{1}-p_{2}+p_{2}-p_{3}+p_{3}-p_{1}=0$ rigorously

Yes, they cancel out: $p-p=\Theta$. Expressions you wrote follow the standard rules of any abelian group. There's nothing more to that.