Homology (mathematics) In mathematics (especially algebraic topologyand abstract algebra), homology (in part fromGreek ὁμός 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, orsingular 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 thehomotopy 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 observation that shapes are distinguished by their 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 defining and categorizing holes in a shape. As it turns out, subtle kinds of holes exist 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 achain complex C(X) encoding information aboutX. A chain complex is a sequence of abelian groups or modules C0, C1, C2, ... connected byhomomorphisms which are called boundary operators. That is,where 0 denotes the trivial group and fori < 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 Cn+ 1 to the group identity in Cn - 1. This means .Now since each Cn is abelian all its subgroups are normal and because and are both subgroups of Cn, is a normal subgroup of and one can consider thefactor groupcalled the n-th homology group of X.We also use the notation and , soComputing 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 Hn(X) of asimplicial 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 Hn(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 nrather than decreasing n; then the groups and follow from the same description andas before.Sometimes, reduced homology groups of a chain complex C(X) are defined as homologies of the augmented complexwherefor a combination Σ niσi of points σi (fixed generators of C0). The reduced homologies coincide with for i≠0. 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 verticesto the sum(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 An to be the free abelian group (or free module) whose generators are all continuous maps from n-dimensional simplices into X. The homomorphisms arise from the boundary maps of simplices.In abstract algebra, one uses homology to definederived 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 moduleF1 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 Fn and homomorphisms pn can be defined. By applying the functor F to this sequence, one obtains a chain complex; the homology Hn of this complex depends only on F and X and is, by definition, then-th derived functor of F, applied to X. Homology functors Chain complexes form a category: A morphism from the chain complex (dn: An → An-1) to the chain complex (en: Bn → Bn-1) is a sequence of homomorphisms fn: An → Bn such that for all n. The n-th homology Hn can be viewed as a covariantfunctor 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 Xbelongs to into the category of abelian groups (or modules).The only difference between homology andcohomology is that in cohomology the chain complexes depend in a contravariant manner onX, 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(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: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 sequenceof chain complexes gives rise to a long exact sequence of homology groupsAll maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps Hn(C) → Hn-1(A) The latter are calledconnecting 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 byHenri Poincaré in his seminal paper "Analysis situs", J. Ecole polytech. (2) 1. 1–121 (1895).The homology group was further developed byEmmy Noether[1][2] and, independently, by Leopold Vietoris and Walther Mayer, in the period 1925–28.[3] Prior to this, topological classes incombinatorial 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
Posted on: Thu, 04 Jul 2013 07:37:00 +0000