Homology (mathematics)

In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space (singular homology) or a group. See homology theory for more background.

For a topological space, the homology groups are generally much easier to compute than the homotopy groups, and consequently one usually will have an easier time working with homology to aid in the classification of spaces.

Construction of homology groups

The procedure works as follows: Given an object such as a topological space $X$, one first defines a chain complex $A=C(X)$ that encodes information about $X$. A chain complex is a sequence of abelian groups or modules $A_{0},A_{1},A_{2},\dots$ connected by homomorphisms $d_{n}:A_{n}\rightarrow A_{n-1}$, such that the composition of any two consecutive maps is zero: $d_{n}\circ d_{n+1}=0$ for all n. This means that the image of the n+1-th map is contained in the kernel of the n-th, and we can define the n-th homology group of X to be the factor group (or quotient module)

$H_{n}(X)=\ker(d_{n})/\mathrm {Im} (d_{n+1})$

The standard notation is $\ker(d_{n})=Z_{n}(X)$ and $\operatorname {im} (d_{n+1})=B_{n}(X)$. Note that the computation of these two groups is usually rather difficult, since they are very large groups. On the other hand, machinery exists that allows one to compute the corresponding homology group easily.

The simplicial homology groups $H_{n}(X)$ of a simplicial complex $X$ are defined using the simplicial chain complex $C(X)$, with $C(X)_{n}$ the free abelian group generated by the $n$-simplices of $X$. The [singular homology]] groups $H_{n}(X)$ are defined for any topological space $X$, and agree with the simplicial homology groups for a simplicial complex.

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 nth map. The homology groups of $X$ therefore measure "how far" the chain complex associated to $X$ is from being exact.

Cohomology groups are formally similar: one starts with a cochain complex, which is the same as a chain complex but whose arrows, now denoted $d^{n}$ point in the direction of increasing n rather than decreasing n; then the groups $\ker(d^{n})=Z^{n}(X)$ and $\operatorname {im} (d^{n-1})=B^{n}(X)$ follow from the same description and

$H^{n}(X)=Z^{n}(X)/B^{n}(X)\$, as before.

Examples

The motivating example comes from algebraic topology: the simplicial homology of a simplicial complex $X$. Here $A_{n}$ is the free abelian group or module whose generators are the n-dimensional oriented simplexes of $X$. The mappings are called the boundary mappings and send the simplex with vertices

$(a[0],a[1],\dots ,a[n])$

to the sum

$\sum _{i=0}^{n}(-1)^{i}(a[0],\dots ,a[i-1],a[i+1],\dots ,a[n]).$

If we take the modules to be over a field, then the dimension of the n-th homology of $X$ turns out to be the number of "holes" in $X$ at dimension n.

Using this example as a model, one can define a singular homology for any topological space $X$. We define a chain complex for $X$ by taking $A_{n}$ to be the free abelian group (or free module) whose generators are all continuous maps from n-dimensional simplices into $X$. The homomorphisms $d_{n}$ arise from the boundary maps of simplices.

In abstract algebra, one uses homology to define derived functors, for example the Tor functors. Here one starts with some covariant additive functor $F$ and some module $X$. The chain complex for $X$ is defined as follows: first find a free module $F_{1}$ and a surjective homomorphism $p_{1}:F_{1}\rightarrow X$. Then one finds a free module $F_{2}$ and a surjective homomorphism $p_{2}:F_{2}\rightarrow \mathrm {ker} (p_{1})$. Continuing in this fashion, a sequence of free modules $F_{n}$ and homomorphisms $p_{n}$ can be defined. By applying the functor $F$ to this sequence, one obtains a chain complex; the homology $H_{n}$ of this complex depends only on $F$ and $X$ and is, by definition, the n-th derived functor of $F$, applied to $X$.

Homology functors

Chain complexes form a category: A morphism from the chain complex $(d_{n}\colon A_{n}\rightarrow A_{n-1})$ to the chain complex $(e_{n}\colon B_{n}\rightarrow B_{n-1})$ is a sequence of homomorphisms $f_{n}\colon A_{n}\rightarrow B_{n}$ such that $f_{n-1}\circ d_{n}=e_{n-1}\circ f_{n}$ for all n. The n-th homology H_n can be viewed as a covariant functor from the category of chain complexes to the category of abelian groups (or modules).

If the chain complex depends on the object X in a covariant manner (meaning that any morphism X → Y induces a morphism from the chain complex of X to the chain complex of Y), then the H_n are covariant functors from the category that X belongs to into the category of abelian groups (or modules).

The only difference between homology and cohomology is that in cohomology the chain complexes depend in a contravariant manner on X, and that therefore the homology groups (which are called cohomology groups in this context and denoted by Hⁿ) form contravariant functors from the category that X belongs to into the category of abelian groups or modules.

Properties

If $(d_{n}:A_{n}\rightarrow A_{n-1})$ is a chain complex such that all but finitely many $A_{n}$ are zero, and the others are finitely generated abelian groups (or finite dimensional vector spaces), then we can define the Euler characteristic

$\chi =\sum (-1)^{n}\,\mathrm {rank} (A_{n})$

(using the rank in the case of abelian groups and the Hamel dimension in the case of vector spaces). It turns out that the Euler characteristic can also be computed on the level of homology:

$\chi =\sum (-1)^{n}\,\mathrm {rank} (H_{n})$

and, especially in algebraic topology, this provides two ways to compute the important invariant $\chi$ for the object $X$ which gave rise to the chain complex.

Every short exact sequence

$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$

of chain complexes gives rise to a long exact sequence of homology groups

$\cdots \rightarrow H_{n}(A)\rightarrow H_{n}(B)\rightarrow H_{n}(C)\rightarrow H_{n-1}(A)\rightarrow H_{n-1}(B)\rightarrow H_{n-1}(C)\rightarrow H_{n-2}(A)\rightarrow \cdots \,$

All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps $H_{n}(C)\rightarrow H_{n-1}(A).$ The latter are called connecting homomorphisms and are provided by the snake lemma.

See also

• Simplicial homology
• Singular homology
• Cohomology
• Homology theory
• Homological algebra

References

• Cartan, Henri Paul and Eilenberg, Samuel (1956) Homological Algebra Princeton University Press, Princeton, NJ, OCLC 529171
• Eilenberg, Samuel and Moore, J. C. (1965) Foundations of relative homological algebra (Memoirs of the American Mathematical Society number 55) American Mathematical Society, Providence, R.I., OCLC 1361982
• Hatcher, A., (2002) Algebraic Topology Cambridge University Press, ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.
• "Homology (Topological space)". PlanetMath.