View source for Notes:Boundary operator

==Definition==
The ''boundary operator'' is a [[group homomorphism]]:
* {{M|1=\partial_p:C_p(K)\rightarrow C_{p-1}(K)}} given by {{M|1=\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]}}
=={{M|p}}-chains==
A ''{{M|p}}-chain'' on {{M|K}} is a function, {{M|c}}, from the set of oriented {{M|p}}-simplicies of {{M|K}} to {{M|\mathbb{Z} }} such that:
# {{M|1=c(\sigma)=-c(\sigma')}} if {{M|\sigma}} and {{M|-\sigma}} are ''opposite'' orientations of the same simplex
# {{M|1=c(\sigma)=0}} for all but finitely many {{M|p}}-simplices, {{M|\sigma}}.
===Elementary chain===
The ''elementary chain'', {{M|c}} corresponding to {{M|\sigma}} is the function defined as:
* {{M|1=c(\sigma)=1}}, {{M|1=c(\sigma')=-1}} for {{M|\sigma'}} being the opposite orientation of {{M|\sigma}} and {{M|1=c(\tau)=0}} for all other simplices, {{M|\tau}}.
We often abuse notation and denote the elementary chain corresponding to {{M|\sigma}} by {{M|\sigma}}.
==Group of (oriented) {{M|p}}-chains==
We can form an (additive) group of {{M|p}}-chains, by simply adding them pointwise. The resulting group is denoted {{M|C_p(K)}}
==The notation {{M|[v_0,\ldots,v_p]}}==
{{M|[v_0,\ldots,v_p]}} denotes the simplex {{M|v_0\ldots v_p}} together with the ordering {{M|(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