Homology (mathematics)

From Wikipedia, the free encyclopedia

In mathematics (especially algebraic topology and abstract algebra), homology (in Greek 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, 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.

[edit] Construction of homology groups

The procedure works as follows. Given an object such as a topological space $X\,$ , one first defines a chain complex $C(X)\,$ encoding information about $X\,$ . A chain complex is a sequence of abelian groups or modules $C_0, C_1, C_2, \dots$ connected by homomorphisms $\partial_n \colon C_n \to C_{n-1},$ which we call boundary operators. That is,

$\dotsb\overset{\partial_{n+1}}{\longrightarrow\,}C_n \overset{\partial_n}{\longrightarrow\,}C_{n-1} \overset{\partial_{n-1}}{\longrightarrow\,} \dotsb \overset{\partial_2}{\longrightarrow\,} C_1 \overset{\partial_1}{\longrightarrow\,} C_0\equiv0,$

where $0\,$ denotes the trivial group and $C_i\equiv0$ for $i\leq0$ . We also require the composition of any two consecutive boundary operators to be zero. That is, for all $n\,$ ,

$\partial_n \circ \partial_{n+1} = 0$ ,

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

Now since each $C_{n}\,$ is abelian, $\operatorname{im}(\partial_{n+1})$ is a normal subgroup of $\ker(\partial_n)$ . And we want to mod out by this subgroup, i.e., consider everything in $\operatorname{im}(\partial_{n+1})$ equivalent and partition $\ker(\partial_n)$ using this equivalence relation. We define the n-th homology group of X to be the factor group (or quotient module)

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

We also use the notation $\ker(\partial_n)=Z_n(X)$ and $\operatorname{im}(\partial_{n+1})=B_n(X)$ , 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 $d n$ 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.

[edit] 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

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

(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 $F 1$ 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 $X$ .

[edit] 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. 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.

[edit] 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 $χ$ 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.

[edit] History

The homology group was 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]

[edit] See also

[edit] Notes

^ Hilton 1988, p. 284
^ For example L'émergence de la notion de group 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, p. 61–63.
^ Bourbaki and Algebraic Topology by John McCleary (PDF) gives documentation (translated into English from French originals).

[edit] 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 60 (5): 282–291, http://www.jstor.org/stable/2689545?origin=JSTOR-pdf
Teicher, M. (ed.) (1999), The Heritage of Emmy Noether, Israel Mathematical Conference Proceedings, Bar-Ilan University/American Mathematical Society/Oxford University Press, OCLC 223099225, ISBN 978-0198510451
Homology (Topological space) on PlanetMath

[0] Hilton 1988, p. 284

[1] For example L'émergence de la notion de group d'homologie, Nicolas Basbois (PDF), in French, note 41, explicitly names Noether as inventing the homology group.

[2] Hirzebruch, Friedrich, "Emmy Noether and Topology" in Teicher 1999, p. 61–63.

[3] Bourbaki and Algebraic Topology by John McCleary (PDF) gives documentation (translated into English from French originals).

[1]

[2]

[3]

[4]

Homology (mathematics)

From Wikipedia, the free encyclopedia

Contents

[edit] Construction of homology groups

[edit] Examples

[edit] Homology functors

[edit] Properties

[edit] History

[edit] See also

[edit] Notes

[edit] References

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