Reduced homology

Builds on Singular homology Relative homology

Definition

Let $X$ be a nonempty topological space with singular chain complex $C_\bullet(X)$ . The augmentation is the homomorphism $\varepsilon \colon C_0(X) \to \Z$ given by $\varepsilon\left(\sum_i n_i \sigma_i\right) = \sum_i n_i$ . The augmented chain complex is $\cdots \to C_2(X) \xrightarrow{\partial_2} C_1(X) \xrightarrow{\partial_1} C_0(X) \xrightarrow{\varepsilon} \Z \to 0,$ with $\Z$ placed in degree $-1$ ; it is a chain complex because $\varepsilon \circ \partial_1 = 0$ . The reduced homology $\tilde H_n(X)$ is the homology of the augmented chain complex in degree $n$ .

Remarks

For $n \ge 1$ , $\tilde H_n(X) = H_n(X)$ , where $H_n$ is singular homology. In degree $0$ , $H_0(X) \cong \tilde H_0(X) \oplus \Z$ , and $\tilde H_{-1}(X) = 0$ because $\varepsilon$ is onto. For any point $x_0 \in X$ , $\tilde H_n(X) \cong H_n(X, \set{x_0})$ for all $n$ , a relative homology group.

Example: for a one-point space $X$ , $C_n(X) \cong \Z$ for every $n \ge 0$ , and $\varepsilon$ is an isomorphism, so $\tilde H_n(X) = 0$ for all $n$ .

Non-example: the hypothesis that $X$ is nonempty matters. For $X = \emptyset$ the map $\varepsilon \colon 0 \to \Z$ is not onto, so the augmented complex gives $\tilde H_{-1}(\emptyset) = \Z$ , and the formula $H_0 \cong \tilde H_0 \oplus \Z$ fails.

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Tag	`SA493EH7DT`
URI	`m2://SA493EH7DT`
Kind	definition
Writing	W0 stub
MSC	55N10
Defines	Reduced homology
Uses	Singular homology, Relative homology
Changed	2 November 2026

Reduced homology

Definition

Remarks

Reading path

Further reading

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