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

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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
A one-vertex framed quiver variety and its Hecke correspondence Definition
Algebraic Lefschetz number via rational homology traces Definition
Augmentation at 0-simplices and reduced singular homology Definition
Borel-Moore homology by locally finite chains Definition
Cover-small singular chains Definition
Euler characteristic of a compact manifold Definition
Kronecker evaluation pairing Definition
Poincare polynomial of a space and of a pair over a field Definition
Relative singular chain complex Definition
Singular cochain complex with coefficients Definition
The reduced Burau homology module Definition
A co-oriented closed transversal detects nonvanishing rational homology of a compact leaf Lemma
Borel-Moore and ordinary homology agree for compact spaces Lemma
First Chern form agrees with the topological line class Lemma
Homology of an infinite CW complex is the colimit of skeletal homology Lemma
Natural singular-cohomology identities are detected on finite regular complexes Lemma
Oriented real two-plane splitting with real-cohomology injection Lemma
Planar facial graph facts and isomorphism extension for arbitrary arc drawings Lemma
Rationalization is exact and commutes with singular homology Lemma
The boundary of a bounded contractible Lipschitz polyhedron is connected Lemma
The kronecker pairing is independent of cocycle and cycle representatives Lemma
The period pairing is well defined and computed by integration Lemma
The rational homology of a closed smooth manifold is finite-dimensional in each degree Lemma
Finiteness and additivity of the Euler characteristic Proposition
Relative homology for the empty and total subspace Proposition
Singular chains and singular homology are covariantly functorial Proposition
The first Hurewicz map is abelianization Proposition
The singular homology of a disjoint union is the direct sum Proposition
The unreduced module fits an exact sequence with the reduced module Proposition
The zero locus of a transverse section represents the Euler dual Proposition
Zero-th singular homology is free on path components Proposition
Reeb-Thurston stability for codimension-one leaves with vanishing first real cohomology Theorem
Singular homology satisfies dimension and arbitrary additivity Theorem
The A1 Nakajima Hecke correspondence gives the fundamental sl2 action Theorem
The symplectic period formula for integrals of wedge products Theorem

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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