What meant by a $k$-chain in a simplicial complex?
The book I'm reading defines $k$-chains as follows:
Let $S$ be a simplicial complex, a $k$-chain is a formal sum $\sum n_i\sigma_i$ where $\sigma_i$ are oriented $k$-simplices and $n_i\in\mathbb{Z}$. For $n_i<0$ define $n_i\sigma_i=(-n_i)(\sigma_i)$ where $-\sigma_i$ is the simplex $\sigma_i$ with negative orientation.
I'm not sure how to interpret this definition, the Wikipedia article for Chains seems to describe a 1-chain as a path between vertices, but what would (using their notation), say, $3t_1+4t_2+t_3$ mean? And how should I interpret $k$-chains for $k>1$?
$\endgroup$ 41 Answer
$\begingroup$Coefficients in a formal sum count the generators of the free abelian group. If $t_1,t_2, ...$ are 1-simplices, then $3t_1+4t_2+t_3$ is a collection of sticks:
- $3$ sticks at $t_1$,
- $4$ sticks at $t_2$,
- and one at $t_3$.
This does not look like a path. (pic (2)). In homology theory, we focus on cycles and seldom use 1-chains that look like $3t_1+4t_2+t_3$.
$-t_1$ means that a stick is placed on $t_1$, but with opposite orientation/direction.
Pic (3) shows that the arithmetic of the formal sums resemble integration paths in calculus, which can be canceled out ($t_1$ and $-t_1$).
