§ Singular homology: induced homomorphism
The space of chains of a topological space
is defined as all functions .
The boundary map is defined as:
where means that we exlude this vertex, and are the vertices
of the domain .
Now, say we have a function , and a singular chain complex for .
In this case, we can induce a chain map , given by:
We wish to show that this produces a homomorphism from
to . To do this, we already have a map from to .
We need to show that it sends and.
The core idea is that if we have abelian groups with subgroups , and a homomorphism
, then this descends to a homomorphism iff
. That is, if whatever is identified in is identified in , then our
morphism will be valid. To prove this, we need to show that if two cosets ,
are equal, then their images under will be equal. We compute ,
and . Since , we get . Thus, the morphism