stylistic:tone
Construction of homology groups

The following text describes a general algorithm for constructing the homology groups. It may be easier for the reader to look at some simple examples first: graph homology and simplicial homology.

The general construction begins with an object such as a topological space X, on which one first defines a chain complex C (X) encoding information about X. A chain complex is a sequence of abelian groups or modules C 0, C 1, C 2, … {\displaystyle C_{0},C_{1},C_{2},\ldots }. connected by homomorphisms ∂ n: C n → C n − 1, {\displaystyle \partial _{n}:C_{n}\to C_{n-1},} which are called boundary operators. That is,

where 0 denotes the trivial group and C i ≡ 0 {\displaystyle C_{i}\equiv 0} for i < 0. It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all n,

i.e., the constant map sending every element of C n + 1 {\displaystyle C_{n+1}} to the group identity in C n − 1. {\displaystyle C_{n-1}.}

The statement that the boundary of a boundary is trivial is equivalent to the statement that i m (∂ n + 1) ⊆ ker (∂ n) {\displaystyle \mathrm {im} (\partial _{n+1})\subseteq \ker(\partial _{n})}, where i m (∂ n + 1) {\displaystyle \mathrm {im} (\partial _{n+1})} denotes the image of the boundary operator and ker (∂ n) {\displaystyle \ker(\partial _{n})} its kernel. Elements of B n (X) = i m (∂ n + 1) {\displaystyle B_{n}(X)=\mathrm {im} (\partial _{n+1})} are called boundaries and elements of Z n (X) = ker (∂ n) {\displaystyle Z_{n}(X)=\ker(\partial _{n})} are called cycles.

Since each chain group C is abelian all its subgroups are normal. Then because ker (∂ n) {\displaystyle \ker(\partial _{n})} is a subgroup of C, ker (∂ n) {\displaystyle \ker(\partial _{n})} is abelian, and since i m (∂ n + 1) ⊆ ker (∂ n) {\displaystyle \mathrm {im} (\partial _{n+1})\subseteq \ker(\partial _{n})} therefore i m (∂ n + 1) {\displaystyle \mathrm {im} (\partial _{n+1})} is a normal subgroup of ker (∂ n) {\displaystyle \ker(\partial _{n})}. Then one can create the quotient group

called the n th homology group of X. The elements of H (X) are called homology classes. Each homology class is an equivalence class over cycles and two cycles in the same homology class are said to be homologous.

A chain complex is said to be exact if the image of the (n +1)th map is always equal to the kernel of the n th map. The homology groups of X therefore measure "how far" the chain complex associated to X is from being exact.

The reduced homology groups of a chain complex C (X) are defined as homologies of the augmented chain complex

where the boundary operator ϵ {\displaystyle \epsilon } is

for a combination ∑ n i σ i, {\displaystyle \sum n_{i}\sigma _{i},} of points σ i, {\displaystyle \sigma _{i},} which are the fixed generators of C. The reduced homology groups H ~ i (X) {\displaystyle {\tilde {H}}_{i}(X)} coincide with H i (X) {\displaystyle H_{i}(X)} for i ≠ 0. {\displaystyle i\neq 0.} The extra Z {\displaystyle \mathbb {Z} } in the chain complex represents the unique map ⟶ X {\displaystyle \longrightarrow X} from the empty simplex to X.

Computing the cycle Z n (X) {\displaystyle Z_{n}(X)} and boundary B n (X) {\displaystyle B_{n}(X)} groups is usually rather difficult since they have a very large number of generators. On the other hand, there are tools which make the task easier.

The simplicial homology groups H (X) of a simplicial complex X are defined using the simplicial chain complex C (X), with C (X) the free abelian group generated by the n -simplices of X. See simplicial homology for details.

The singular homology groups H (X) are defined for any topological space X, and agree with the simplicial homology groups for a simplicial complex.

Cohomology groups are formally similar to homology groups: one starts with a cochain complex, which is the same as a chain complex but whose arrows, now denoted d n, {\displaystyle d_{n},} point in the direction of increasing n rather than decreasing n; then the groups ker (d n) = Z n (X) {\displaystyle \ker \left(d^{n}\right)=Z^{n}(X)} of cocycles and i m (d n − 1) = B n (X) {\displaystyle \mathrm {im} \left(d^{n-1}\right)=B^{n}(X)} of coboundaries follow from the same description. The n th cohomology group of X is then the quotient group

in analogy with the n th homology group.