In general, the homology of a point is not trivial but is concentrated in degree 0 on the given coefficient object. For some applications, though, it is convenient to divide out that contribution such as to have the homology of the point be entirely trivial. This is called reduced homology.
Reduced singular homology
We discuss the reduced version of singular homology.
Let be a topological space. Write for its singular chain complex.
The augmentation map is the homomorphism of abelian groups
which adds up all the coefficients of all 0-chains:
Since the boundary of a 1-chain is in the kernel of this map, it constitutes a chain map
where now is regarded as a chain complex concentrated in degree 0.
The reduced singular chain complex of is the kernel of the augmentation map, the chain complex sitting in the short exact sequence
The reduced singular homology of is the chain homology of the reduced singular chain complex
The reduced singular homology of , denoted , is the chain homology of the augmented chain complex
Relation to ordinary homology
Let be a topological space, its singular homology and its reduced singular homology, def. 2.
For there is an isomorphism
The homology long exact sequence of the defining short exact sequence is, since here is concentrated in degree 0, of the form
Here exactness says that all the morphisms for positive are isomorphisms. Moreover, since is a free abelian group, hence a projective object, the remaining short exact sequence
is split (as discussed there) and hence .
For the point, the morphism
is an isomorphism. Accordingly the reduced homology of the point vanishes in every degree:
By the discussion at Singular homology – Relation to homotopy groups we have that
Moreover, it is clear that is the identity? map.
Relation to relative homology
Consider the sequence of topological subspace inclusions
By the discussion at Relative homology - long exact sequences this induces a long exact sequence of the form
Here in positive degrees we have and therefore exactness gives isomorphisms
and hence with prop. 1 isomorphisms
It remains to deal with the case in degree 0. To that end, observe that is a monomorphism: for this notice that we have a commuting diagram
where is the terminal map. That the outer square commutes means that and hence the composite on the left is an isomorphism. This implies that is an injection.
Therefore we have a short exact sequence as shown in the top of this diagram
Using this we finally compute
Relation to wedge sums
Let be a set of pointed topological spaces. Write for their wedge sum and write for the canonical inclusion functions.
For each the homomorphism
is an isomorphism.
For instance (Hatcher, cor. 2.25).
For singular homology
For a topological space, write for its singular homology with integer coefficients.
The reduced singular homology of the 0-sphere is
Reduced singular homology is discussed for instance around p. 119 of