DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-05

How statement and proof provenance work

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The singular chain complex and singular homology

Definition

For a topological space $X$ and an abelian group $G$ , the singular chain groups and boundary maps of The singular boundary operator form the singular chain complex $C_{∙} (X; G) : = (\dots \to C_{n} (X; G) \overset{\partial_{n}}{\to} C_{n - 1} (X; G) \to \dots),$ because The singular boundary squares to zero gives $\partial_{n - 1} \partial_{n} = 0$ .

Its degree- $n$ cycles and boundaries are $Z_{n}^{s i n g} (X; G) : = \ker \partial_{n}, B_{n}^{s i n g} (X; G) : = im \partial_{n + 1},$ in the sense of Cycle and boundary subobjects of a complex.

The $n$ th singular homology group is the homology object of this chain complex: $H_{n}^{s i n g} (X; G) : = Z_{n}^{s i n g} (X; G) / B_{n}^{s i n g} (X; G),$ equivalently $H_{n} (C_{∙} (X; G))$ in the notation of Homology object of a chain complex. When the coefficient group is $Z$ , write simply $C_{n} (X)$ and $H_{n} (X)$ when no confusion can arise.

Depends on

Used by

Homotopic maps induce the same map on singular homology Corollary
Equal homology does not imply homotopy equivalence Counterexample
Augmentation at 0-simplices and reduced singular homology Definition
Singular chains and singular homology are covariantly functorial Proposition
The singular homology of a disjoint union is the direct sum Proposition
Zero-th singular homology is free on path components Proposition

Dependency tree · two levels

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Sources

Allen Hatcher, Algebraic Topology (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology (standard reference, not scraped)