Singular Homology Groups

Core Idea

The singular homology groups H_n(X) = ker(d_n)/im(d_{n+1}) of the singular chain complex are the fundamental topological invariants of algebraic topology. They are defined for any topological space, are invariant under homotopy equivalence (not just homeomorphism), and are functorial — continuous maps induce homomorphisms on homology. For spaces admitting triangulations, singular homology agrees with simplicial homology, but singular homology's true power lies in the theoretical tools (long exact sequences, excision, Mayer-Vietoris) that make it computable without ever writing down the enormous chain groups explicitly.

Explainer

Having defined the singular chain complex C_*(X) with its boundary operators d_n, the singular homology groups H_n(X) = ker(d_n)/im(d_{n+1}) are the central objects of algebraic topology. Every topological space has singular homology groups, and these groups are computable (for reasonable spaces) despite the apparently intractable size of the chain groups. The key properties that make singular homology powerful are homotopy invariance, functoriality, and a suite of computational tools (long exact sequences, excision, Mayer-Vietoris) that allow indirect computation.

Functoriality means that a continuous map f : X -> Y induces group homomorphisms f_* : H_n(X) -> H_n(Y) for all n, defined by f_*([\alpha]) = [f compose alpha] on chains. This respects composition: (g compose f)_* = g_* compose f_*, and id_* = id. In categorical language, H_n is a functor from the category of topological spaces to the category of abelian groups. The practical consequence is that any topological relationship between spaces (inclusion, retraction, covering map, quotient map) translates into an algebraic relationship between their homology groups. This is the fundamental strategy of algebraic topology: convert topological questions into algebraic ones, which are often easier.

Homotopy invariance is the theorem that homotopic maps f, g : X -> Y induce the same homomorphism on homology: f_* = g_*. The proof constructs a chain homotopy — a sequence of homomorphisms P_n : C_n(X) -> C_{n+1}(Y) satisfying f_# - g_# = d compose P + P compose d — using the prism operator, which triangulates the product Deltaⁿ x [0,1] and maps it into Y using the homotopy. The chain homotopy equation implies that f_# and g_# induce identical maps on homology (their difference sends every cycle to a boundary). As an immediate corollary, homotopy equivalent spaces have isomorphic homology: if f : X -> Y is a homotopy equivalence with homotopy inverse g, then f_* and g_* are mutually inverse isomorphisms.

For basic spaces: a point has H_0 = Z and H_n = 0 for n > 0. Any contractible space (Rⁿ, Dⁿ, star-shaped regions) has the same homology, by homotopy invariance. The circle S¹ has H_0 = Z, H_1 = Z, and H_n = 0 for n >= 2. More generally, the n-sphere Sⁿ has H_0 = Z, H_n = Z, and all other homology zero. The torus T² has H_0 = Z, H_1 = Z², H_2 = Z. These basic computations, which we will establish using the tools developed in subsequent topics, serve as the building blocks from which the homology of more complex spaces is assembled via exact sequences and excision.

The agreement between singular and simplicial homology (for triangulable spaces) is a nontrivial theorem. The proof proceeds by showing that the inclusion of the simplicial chain complex into the singular chain complex induces isomorphisms on homology. This justifies the simplicial approach to computation: for a space with a known triangulation, compute with the small simplicial chain groups and obtain the same answer as the enormous singular chain complex would give. But singular homology's universality and homotopy invariance make it the correct theoretical framework: it is singular homology that satisfies the Eilenberg-Steenrod axioms that characterize homology theories, and all the major theorems (Mayer-Vietoris, excision, universal coefficients, Poincare duality) are most naturally stated and proved for singular homology.