Define the **reduced homology groups** to be the homology groups of the augmented chain complex where . We require to be nonempty, to avoid having a nontrivial homology group in dimension -1.

**Relation between and **

Since , vanishes on and hence induces a map with . So . Clearly, for .

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Intuitively what does reduced homology mean ?

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Same as homology, except for degree 0. For a point, reduced homology is zero for all degree.

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