Reduced Homology

Reduced Homology and Path Components

Recall that the singular homology of a point is:

The homology groups of a point are:

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It is often convenient to modify the definition of homology so that a point has trivial homology in all dimensions. This leads to reduced homology.

Reduced Homology

The reduced singular homology groups, denoted , are the homology groups of the augmented chain complex: where the augmentation map is defined by .

Remark

This definition makes sense, since can be recognized as the empty set, and there is a unique map from the empty set to any space, so .

e.g. For a point, the augmentation map is an isomorphism, which makes the augmented complex exact. Thus the reduced homology of a point is trivial in all dimensions:

Proposition

Let be a non-empty, path-connected space. Then and .

Proof The map is surjective since . The first isomorphism theorem tells that iff , so it’s sufficient to show . From the above diagram, we know that the reduced homology is the kernel of the induced map . Let , where is a 0-cycle. The condition for to be in the kernel is that . Since is path-connected, for any two points , there exists a path from to . This path is a 1-simplex, and its boundary is . Because , we can express as a sum of terms like , and each of these is the boundary of a path from to . Therefore, is a boundary, which means in . Thus, .

Homology of Disjoint Unions

Direct Sum of Chain Complexes

If and are chain complexes, then their direct sum is the chain complex with , and the boundary map defined by .

Lemma

The following properties hold:

• ,
• , moreover, .

Corollary

If a space is a disjoint union of its path components , then the homology groups are direct sums: In particular, for any space , counts the number of path-components of .