Notes:Boundary operator

Definition

The boundary operator is a group homomorphism:

• [ilmath]\partial_p:C_p(K)\rightarrow C_{p-1}(K)[/ilmath] given by [ilmath]\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][/ilmath]

[ilmath]p[/ilmath]-chains

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

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

Elementary chain

The elementary chain, [ilmath]c[/ilmath] corresponding to [ilmath]\sigma[/ilmath] is the function defined as:

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

We often abuse notation and denote the elementary chain corresponding to [ilmath]\sigma[/ilmath] by [ilmath]\sigma[/ilmath].

Group of (oriented) [ilmath]p[/ilmath]-chains

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

The notation [ilmath][v_0,\ldots,v_p][/ilmath]

[ilmath][v_0,\ldots,v_p][/ilmath] denotes the simplex [ilmath]v_0\ldots v_p[/ilmath] together with the ordering [ilmath](v_0,\ldots,v_p)[/ilmath] 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