Notes:Boundary operator

Definition

The boundary operator is a group homomorphism:

• $\partial_p:C_p(K)\rightarrow C_{p-1}(K)$ given by $\partial_p\sigma:=\partial_p[v_0,\ldots,v_p]=\sum^p_{i=0}(-1)^i[v_0,...,v_{i-1},v_{i+1},\ldots,v_p]$

$p$-chains

A $p$-chain on $K$ is a function, $c$, from the set of oriented $p$-simplicies of $K$ to $\mathbb{Z}$ such that:

1. $c(\sigma)=-c(\sigma')$ if $\sigma$ and $-\sigma$ are opposite orientations of the same simplex
2. $c(\sigma)=0$ for all but finitely many $p$-simplices, $\sigma$.

Elementary chain

The elementary chain, $c$ corresponding to $\sigma$ is the function defined as:

• $c(\sigma)=1$, $c(\sigma')=-1$ for $\sigma'$ being the opposite orientation of $\sigma$ and $c(\tau)=0$ for all other simplices, $\tau$.

We often abuse notation and denote the elementary chain corresponding to $\sigma$ by $\sigma$.

Group of (oriented) $p$-chains

We can form an (additive) group of $p$-chains, by simply adding them pointwise. The resulting group is denoted $C_p(K)$

The notation $[v_0,\ldots,v_p]$

$[v_0,\ldots,v_p]$ denotes the simplex $v_0\ldots v_p$ together with the ordering $(v_0,\ldots,v_p)$ of its vertices AND their equivalence classes.

Recall that two orderings are considered equivalent if they differ by an even permutation of the vertices.

Source

Munkres - Elements of Algebraic Topology