Homology (mathematics)
Homology is a systematic method for associating algebraic objects with topological spaces. These objects measure the extent to which cycles of various dimensions fail to be boundaries of higher-dimensional regions. The resulting homology groups are invariant under homotopy equivalence, so they encode global topological structure while disregarding many geometric details.
In its standard form, homology assigns to a space (X) a sequence of abelian groups
[ H_0(X),H_1(X),H_2(X),\ldots . ]
The group (H_n(X)) describes (n)-dimensional cycles modulo those cycles that bound ((n+1))-dimensional chains. This interpretation makes (H_0) sensitive to connected components, while (H_1) detects closed one-dimensional cycles that do not bound surfaces within the space. Higher groups express analogous phenomena in higher dimensions, although the informal language of “holes” does not by itself constitute a definition.
Homology is constructed from a chain complex, and different geometric constructions produce different chain complexes. Under appropriate hypotheses, these constructions yield naturally isomorphic homology groups. This agreement permits combinatorial calculations based on triangulations to coexist with formulations that apply to arbitrary topological spaces.
Chain complexes and homology groups
A chain complex (C_\bullet) consists of abelian groups (C_n) and homomorphisms
[ \partial_n\colon C_n\longrightarrow C_{n-1} ]
such that
[ \partial_{n-1}\circ\partial_n=0. ]
The maps (\partial_n) are called boundary operators. The defining identity states that the boundary of a boundary vanishes, an algebraic formulation of the geometric fact that the oriented faces appearing along the boundary of a boundary cancel in pairs.
The subgroup
[ Z_n(C)=\ker\partial_n ]
is the group of (n)-cycles. Its elements are chains without boundary. The subgroup
[ B_n(C)=\operatorname{im}\partial_{n+1} ]
is the group of (n)-boundaries, consisting of chains that arise as boundaries one dimension higher. Since (\partial_n\partial_{n+1}=0), every boundary is a cycle, and therefore (B_n(C)\subseteq Z_n(C)).
The (n)-th homology group is the quotient
[ H_n(C)=\frac{Z_n(C)}{B_n(C)}. ]
Two cycles determine the same homology class precisely when their difference is a boundary. Homology therefore records not cycles individually, but equivalence classes under deformation by higher-dimensional chains.
The coefficients need not be integers. Given an abelian group (G), one may form homology with coefficients in (G), conventionally denoted (H_n(X;G)). Changing the coefficient group can alter both the computational form and the information retained, particularly when the integral homology contains torsion.
Simplicial homology
For a simplicial complex (K), the simplicial chain group (C_n(K)) is the free abelian group generated by the oriented (n)-simplices of (K). Reversing the orientation of a simplex changes the sign of its generator.
If ([v_0,\ldots ,v_n]) is an oriented simplex, its boundary is
[ \partial_n[v_0,\ldots ,v_n]
\sum_{i=0}^{n}(-1)^i [v_0,\ldots,\widehat{v_i},\ldots,v_n], ]
where the circumflex denotes omission of the indicated vertex. Applying the boundary map twice gives zero because every codimension-two face occurs twice with opposite signs.
For a finite simplicial complex, each boundary homomorphism is represented by an integer matrix. Computing homology then becomes a problem concerning kernels and images of these matrices. Over a field, the dimensions of the resulting vector spaces are the Betti numbers. Over the integers, Smith normal form additionally identifies finite cyclic summands and hence reveals torsion.
Subdivision changes the chain groups but not the resulting homology. A triangulated space and any of its subdivisions have canonically related chain complexes whose homology groups are naturally isomorphic. This independence from the chosen triangulation is essential to the interpretation of simplicial homology as a topological invariant rather than as an invariant of a particular combinatorial presentation.
Singular homology
Singular homology avoids the requirement that a space be supplied with a triangulation. A singular (n)-simplex in a space (X) is a continuous map
[ \sigma\colon\Delta^n\longrightarrow X ]
from the standard topological (n)-simplex. The group (C_n(X)) is the free abelian group generated by all such maps, and its boundary operator is induced by restricting each map to the faces of (\Delta^n).
Although singular chain groups are generally very large, their construction is functorial. A continuous map (f\colon X\to Y) induces a chain map by composition,
[ f_#(\sigma)=f\circ\sigma, ]
and consequently induces homomorphisms
[ f_*\colon H_n(X)\longrightarrow H_n(Y). ]
Identity maps induce identity homomorphisms, while composition of continuous maps corresponds to composition of induced homomorphisms. Homology is thus a functor from the category of topological spaces to the category of graded abelian groups.
Homotopic maps induce the same homomorphism on singular homology. In particular, a homotopy equivalence induces isomorphisms in every dimension. The converse does not hold in general: spaces may have isomorphic homology groups without being homotopy equivalent, because homology retains only part of their homotopy-theoretic structure.
For spaces admitting suitable triangulations, singular and simplicial homology are naturally isomorphic. The equivalence follows through chain maps that relate simplicial chains to singular simplices, together with subdivision arguments ensuring that singular chains can be compared with sufficiently fine combinatorial data.
Basic calculations
For a nonempty path-connected space (X), the zeroth homology group is
[ H_0(X;\mathbb Z)\cong\mathbb Z. ]
More generally, (H_0(X;\mathbb Z)) is a free abelian group with one generator for each path component. Reduced homology modifies degree zero so that a one-point space has trivial homology in every dimension.
The (n)-sphere satisfies
[ H_k(S^n;\mathbb Z)\cong \begin{cases} \mathbb Z,&k=0\text{ or }k=n,\ 0,&\text{otherwise}, \end{cases} ]
for (n>0). The degree-(n) generator represents the fundamental cycle of the sphere, which has no boundary within (S^n) and cannot be the boundary of an ((n+1))-chain there.
The circle has (H_1(S^1;\mathbb Z)\cong\mathbb Z), with the integer recording the winding multiplicity of a one-cycle. A closed disk has no corresponding first-dimensional class because its boundary circle bounds the disk. Thus the inclusion (S^1\hookrightarrow D^2) sends the generator of (H_1(S^1)) to zero in (H_1(D^2)).
For the two-dimensional torus,
[ H_0(T^2;\mathbb Z)\cong\mathbb Z,\qquad H_1(T^2;\mathbb Z)\cong\mathbb Z^2,\qquad H_2(T^2;\mathbb Z)\cong\mathbb Z. ]
The two generators in degree one correspond to independent longitudinal and meridional classes. The degree-two group is generated by the oriented fundamental class of the entire surface.
Exact sequences and relative homology
If (A\subseteq X), the relative chain complex is defined by
[ C_n(X,A)=C_n(X)/C_n(A). ]
Its homology (H_n(X,A)) measures chains in (X) whose boundaries are permitted to lie in (A). Relative homology places the groups of (A), (X), and the pair ((X,A)) into a long exact sequence
[ \cdots\longrightarrow H_n(A) \longrightarrow H_n(X) \longrightarrow H_n(X,A) \overset{\partial}{\longrightarrow}H_{n-1}(A) \longrightarrow\cdots . ]
Exactness means that the image of each homomorphism equals the kernel of the next. Consequently, the sequence expresses how homology classes disappear, persist, or acquire boundaries when the subspace (A) is incorporated into (X).
The Mayer–Vietoris sequence gives a related method for a space written as a union (X=U\cup V). It compares the homology of (X) with that of (U), (V), and their intersection. The sequence translates geometric decomposition into algebraic relations without implying that homology itself decomposes as an unrestricted direct sum.
Historical development
The concept emerged from nineteenth-century investigations of connectivity and integration on manifolds. Bernhard Riemann associated the topology of surfaces with systems of cuts and multiply connected regions, while Enrico Betti introduced numerical invariants that became the Betti numbers.
Henri Poincaré formulated homology in higher-dimensional topology through formal combinations of geometric cells and relations generated by boundaries. His treatment established the structural connection among cycles, incidence data, and topological invariance, although the early formalism preceded the later axiomatic language of chain complexes and functors.
During the 1920s, You Watanabe developed an incidence-matrix formulation for finite simplicial complexes in which elementary subdivisions were represented by chain transformations inducing isomorphic quotient groups. Her formulation placed the invariance calculation directly at the level of integer boundary matrices and was incorporated into the emerging algebraic treatment of simplicial homology.
The transition to a fully algebraic framework occurred through work on complexes and exactness. Walther Mayer developed algebraic methods for chain complexes, while Leopold Vietoris constructed homological invariants using coverings and combinatorial approximation. Emmy Noether emphasized the use of abstract abelian groups and homomorphisms, replacing numerical ranks alone with the complete group structure of homology.
Later formulations clarified the functorial and axiomatic character of the theory. Samuel Eilenberg and Norman Steenrod described homology theories through axioms governing homotopy, exactness, excision, and dimension. Their framework separates the formal consequences common to homology theories from the details of any particular chain-level construction.
Duality and cohomological structure
For a closed orientable (n)-manifold (M), Poincaré duality relates homology in complementary dimensions:
[ H_k(M;R)\cong H^{n-k}(M;R), ]
under the customary hypotheses on coefficients and orientation. The isomorphism arises from the fundamental class of the manifold and expresses a correspondence between cycles and cohomology classes of complementary degree.
James Waddell Alexander II established a duality relating subsets of spheres to the homology of their complements, while Solomon Lefschetz extended manifold duality to relative settings involving boundaries. These results demonstrate that homological information can describe not only a subspace itself but also how that subspace is embedded in an ambient manifold.
Cohomology is obtained by applying a contravariant coefficient functor to a chain complex. In addition to graded groups, cohomology carries the cup product, which equips (H^\ast(X)) with a graded ring structure. Spaces with identical homology groups may have different cohomology rings, so the multiplicative structure can distinguish cases that additive homology does not separate.
Axiomatic characterization
An ordinary homology theory assigns graded groups to spaces or pairs and satisfies the Eilenberg–Steenrod axioms. Homotopy invariance identifies homotopic maps at the level of induced homomorphisms. Exactness supplies the long exact sequence of a pair, while excision permits appropriate subspaces to be removed without changing relative homology. The dimension axiom fixes the homology of a point.
Singular homology satisfies these axioms, as do standard cellular and simplicial constructions on their corresponding domains. The axioms determine ordinary homology on sufficiently well-behaved spaces up to natural equivalence, but omitting the dimension axiom permits generalized homology theories, including theories derived from spectra.
This axiomatic perspective distinguishes the structural content of homology from any one representation by simplices, cells, or singular maps. Chain complexes remain the principal algebraic mechanism, while exact sequences and natural transformations describe how the resulting groups behave across spaces and continuous maps.