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 or a group. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly In Mathematics, a sequence is an ordered list of objects (or events An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element 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. In Mathematics, homology theory is the Axiomatic study of the intuitive geometric idea of homology of cycles on Topological spaces It can be broadly In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space In Abstract algebra, Homological algebra, Algebraic topology and Algebraic number theory, as well as in applications to Group theory proper

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. In Mathematics, homotopy groups are used in Algebraic topology to classify Topological spaces The base point preserving maps from an n -dimensional

1 Construction of homology groups
2 Examples
3 Homology functors
4 Properties
5 See also
6 References

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 $X$ . In Mathematics, a chain complex is a construct originally used in the field of Algebraic topology. 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. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function 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 $\operatorname{im}(d_{n+1})=B_n(X)$ . In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Abstract algebra, a branch of Mathematics, given a module and a Submodule, one can construct their quotient module. 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 $X$ . In Mathematics, in the area of Algebraic topology, simplicial homology is a theory with a Finitary definition and is probably the most tangible variant In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments 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. In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space

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. In Mathematics, a chain complex is a construct originally used in the field of Algebraic topology.

Examples

The motivating example comes from algebraic topology: the simplicial homology of a simplicial complex $X$ . Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Mathematics, in the area of Algebraic topology, simplicial homology is a theory with a Finitary definition and is probably the most tangible variant In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments Here $A n$ is the free abelian group or module whose generators are the n-dimensional oriented simplexes of $X$ . In Abstract algebra, a free abelian group is an Abelian group that has a "basis" in the sense that every element of the group can be written in 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 $X$ . Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. 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 $X$ . In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Geometry, a simplex (plural simplexes or simplices) or n -simplex is an n -dimensional analogue of a triangle 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. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, certain Functors may be derived to obtain other functors closely related to the original ones In higher Mathematics, the Tor functors of Homological algebra are the Derived functors of the Tensor product functor 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 $p_1 : F_1 \rightarrow X$ . In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every 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 $f_{n-1} \circ d_n = e_{n} \circ f_n$ for all n. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets 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). In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories

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). In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories

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. In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain

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). In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant In Mathematics, the rank, or torsion-free rank, of an Abelian group measures how large a group is in terms of how large a Vector space over the In Mathematics, the dimension of a Vector space V is the cardinality (i 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 $H_n(C) \rightarrow H_{n-1}(A).$ The latter are called connecting homomorphisms and are provided by the snake lemma. In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group In Mathematics, particularly Homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the

References

Cartan, Henri Paul and Eilenberg, Samuel (1956) Homological Algebra Princeton University Press, Princeton, NJ, OCLC 529171
Eilenberg, Samuel and Moore, J. In Mathematics, in the area of Algebraic topology, simplicial homology is a theory with a Finitary definition and is probably the most tangible variant In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space In Mathematics, homology theory is the Axiomatic study of the intuitive geometric idea of homology of cycles on Topological spaces It can be broadly Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting 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.
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Contents

Construction of homology groups

Examples

Homology functors

Properties

See also

References