Singular homology and homotopy invariance

For a space X and fixed commutative ring R, the singular chain module $C_n(X;R)$ is free on all continuous maps $\sigma:\Delta^n\to X$. These maps need not be embeddings and may repeat vertices or collapse an entire simplex to a point. Restricting a map to its faces and taking alternating signs defines the boundary. The same face-pair cancellation as in simplicial chains gives boundary squared zero. A chain remains a finite formal combination even when the set of all possible generators is infinite.

A continuous map from X to Y sends each singular simplex to its composite with that map. Composition commutes with taking faces, hence gives a chain map and therefore a homomorphism on each homology group. Identity maps induce identity homomorphisms, and composition of maps induces composition of homomorphisms. These properties are functoriality. They distinguish the construction from merely assigning a list of groups to each space: the induced maps are essential for exact sequences, degree calculations, and comparing inclusions.

Suppose continuous maps f and g are joined by a continuous homotopy $F:X\times[0,1]\to Y$. Sweeping an n-simplex through the interval gives a prism. Subdividing that prism into oriented simplices constructs maps $P_n:C_n(X)\to C_{n+1}(Y)$ with $g_\#-f_\#=\partial P+P\partial$, using compatible orientations. This is a chain homotopy. The geometric construction and its signs justify the identity; merely saying that nearby maps should have the same holes does not prove it.

For a cycle z the second term vanishes, so the difference between its two images is a boundary. Thus homotopic maps induce exactly the same homomorphism on homology. If two spaces are homotopy equivalent, applying this fact to the two composites proves that their induced maps are mutually inverse. There is no requirement of compactness, a manifold structure, or an embedding in Euclidean space for this singular-homology theorem. The maps and homotopy must, however, be continuous on their full stated domains.

A nonempty contractible space has the homology of a point: ordinary degree zero is the coefficient ring and all positive degrees vanish. Its singular chain groups do not vanish in positive degrees; there are continuous simplices into even a single point in every degree. A chain contraction or direct calculation shows why the homology vanishes despite those generators. Confusing the chain groups with the final quotient would produce an incorrect point example and an incorrect interpretation of contractibility.

The converse implication is false: vanishing reduced homology alone does not guarantee contractibility for arbitrary spaces. Homology also abelianizes some information that the fundamental group retains. For a path-connected space, integer H₁ is the abelianization of its fundamental group, so noncommuting loops can become indistinguishable after passage to H₁. Deformation retracts are convenient computational tools because they provide explicit homotopy equivalences, but a picture suggesting a retraction needs a continuous map that behaves correctly at every boundary and identified point.