Homology (mathematics)

In mathematics (especially algebraic topology and abstract algebra), homology (in part from Greek ὁμόιος 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 or a group. See homology theory for more background, or singular homology for a concrete version for topological spaces, or group cohomology for a concrete version for groups.

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.

The original motivation for defining homology groups is the commonplace observation that one aspect of the shape of an object is its holes. But because a hole is "not there", it is not immediately obvious how to define a hole, or how to distinguish between different kinds of holes. Homology is a rigorous mathematical method for detecting and categorizing holes in a shape. As it turns out, there exist subtle kinds of holes that homology cannot "see" — in which case homotopy groups may be what is needed.

Construction of homology groups

The 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 C₀, C₁, C₂, ... connected by homomorphisms \( \partial_n \colon C_n \to C_{n-1} \) , which are called boundary operators. That is,

\( \dotsb\overset{\partial_{n+1}}{\longrightarrow\,}C_n \overset{\partial_n}{\longrightarrow\,}C_{n-1} \overset{\partial_{n-1}}{\longrightarrow\,} \dotsb \overset{\partial_2}{\longrightarrow\,} C_1 \overset{\partial_1}{\longrightarrow\,} C_0\overset{\partial_0}{\longrightarrow\,} 0 \)

where 0 denotes the trivial group and \( C_i\equiv0 \) for i < 0. It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all n,

\( \partial_n \circ \partial_{n+1} = 0_{n+1,n-1}, \, \)

i.e., the constant map sending every element of C_{n + 1} to the group identity in C_{n - 1}. This means \mathrm{im}(\partial_{n+1})\subseteq\ker(\partial_n) \) .

Now since each Cn is abelian all its subgroups are normal and because \( \mathrm{im}(\partial_{n+1}) and \ker(\partial_n) are both subgroups of \( Cn, \mathrm{im}(\partial_{n+1}) \) is a normal subgroup of \( \ker(\partial_n) \) and one can consider the factor group

\( H_n(X) := \ker(\partial_n) / \mathrm{im}(\partial_{n+1}), \, \)

called the n-th homology group of X.

We also use the notation \( \ker(\partial_n)=Z_n(X) \) and \( \mathrm{im}(\partial_{n+1})=B_n(X), \) so

\( H_n(X)=Z_n(X)/B_n(X). \,

Computing these two groups is usually rather difficult since they are very large groups. On the other hand, we do have tools which make the task easier.

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 dⁿ point in the direction of increasing n rather than decreasing n; then the groups

\( \ker(d^n) = Z^n(X) \) and \( \mathrm{im}(d^{n + 1}) = B^n(X) \) 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 An 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

\( \sum_{i=0}^n (-1)^i(a[0], \dots, a[i-1], a[i+1], \dots, a[n]) \)

(which is considered 0 if n = 0).

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 \( \partial_n \) 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 F₁ and a surjective homomorphism p₁: F₁ → X. Then one finds a free module F₂ and a surjective homomorphism p₂: F₂ → ker(p₁). 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

Homology functors

Chain complexes form a category: A morphism from the chain complex (d_n: A_n → A_n-1) to the chain complex (e_n: B_n → B_n-1) is a sequence of homomorphisms f_n: A_n → B_n such that \( f_{n-1} \circ d_n = e_{n} \circ f_n \) for all n. The n-th homology Hn 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 Hn 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 (dn: An → An-1) is a chain complex such that all but finitely many An 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 χ 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) → H_n-1(A) The latter are called connecting homomorphisms and are provided by the snake lemma. The snake lemma can be applied to homology in numerous ways that aid in calculating homology groups, such as the theories of relative homology and Mayer-Vietoris sequences.

History

Homology classes were first defined rigorously by Henri Poincaré in his seminal paper "Analysis situs", J. Ecole polytech. (2) 1. 1–121 (1895).

The homology group was further developed by Emmy Noether[1][2] and, independently, by Leopold Vietoris and Walther Mayer, in the period 1925–28.[3] Prior to this, topological classes in combinatorial topology were not formally considered as abelian groups. The spread of homology groups marked the change of terminology and viewpoint from "combinatorial topology" to "algebraic topology".[4]
See also

Simplicial homology
Singular homology
Cellular homology
Homology theory
Homological algebra
Cohomology

Notes

^ Hilton 1988, p. 284
^ For example L'émergence de la notion de groupe d'homologie, Nicolas Basbois (PDF), in French, note 41, explicitly names Noether as inventing the homology group.
^ Hirzebruch, Friedrich, "Emmy Noether and Topology" in Teicher 1999, pp. 61–63.
^ Bourbaki and Algebraic Topology by John McCleary (PDF) gives documentation (translated into English from French originals).

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.
Hilton, Peter (1988), "A Brief, Subjective History of Homology and Homotopy Theory in This Century", Mathematics Magazine (Mathematical Association of America) 60 (5): 282–291, JSTOR 2689545
Teicher, M. (ed.) (1999), The Heritage of Emmy Noether, Israel Mathematical Conference Proceedings, Bar-Ilan University/American Mathematical Society/Oxford University Press, ISBN 978-0-19-851045-1, OCLC 223099225
Homology (Topological space) on PlanetMath

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