|
|
| (One intermediate revision by the same user not shown) |
| Line 1: |
Line 1: |
| In [[mathematics]] (especially [[algebraic topology]] and [[abstract algebra]]), '''homology''' (in part from [[Greek language|Greek]] ὁμός ''homos'' "identical") is a certain general procedure to associate a [[sequence]] of [[abelian group]]s or [[module (mathematics)|modules]] with a given mathematical object such as a [[topological space]] or a [[group (mathematics)|group]]. See [[singular homology]] for a concrete version for topological spaces, or [[group cohomology]] for a concrete version for groups.
| | She is known by the title of Myrtle Shryock. To gather cash is one of the things I love most. California is our beginning location. Hiring is her day job now but she's usually needed her personal company.<br><br>Also visit my homepage; [http://Tvc.in/mealdeliveryservice77873 Tvc.in] |
| | |
| For a [[topological]] space, the homology groups are generally much easier to compute than the [[homotopy group]]s, 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 a ''[[chain complex]]'' ''C(X)'' encoding information about ''X''. A chain complex is a sequence of abelian groups or modules ''C''<sub>0</sub>, ''C''<sub>1</sub>, ''C''<sub>2</sub>, ... connected by [[group homomorphism|homomorphisms]] <math> \partial_n \colon C_n \to C_{n-1},</math> which are called '''boundary operators'''. That is,
| |
| | |
| :<math>\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</math>
| |
| | |
| where 0 denotes the trivial group and <math>C_i\equiv0</math> for ''i'' < 0. It is also required that the composition of any two consecutive boundary operators be trivial. That is, for all ''n'',
| |
| | |
| :<math> \partial_n \circ \partial_{n+1} = 0_{n+1,n-1}, \, </math>
| |
| | |
| i.e., the constant map sending every element of ''C''<sub>''n'' + 1</sub> to the group identity in ''C''<sub>''n'' - 1</sub>. This means <math>\mathrm{im}(\partial_{n+1})\subseteq\ker(\partial_n)</math>.
| |
| | |
| Now since each ''C<sub>n</sub>'' is abelian all its subgroups are normal and because <math>\mathrm{im}(\partial_{n+1})</math> and <math>\ker(\partial_n)</math> are both subgroups of ''C<sub>n</sub>'', <math>\mathrm{im}(\partial_{n+1})</math> is a [[normal subgroup]] of <math>\ker(\partial_n)</math> and one can consider the [[factor group]]
| |
| :<math> H_n(X) := \ker(\partial_n) / \mathrm{im}(\partial_{n+1}), \, </math>
| |
| called the '''''n''-th homology group of ''X'''''.
| |
| | |
| We also use the notation <math>\ker(\partial_n)=Z_n(X) </math> and <math>\mathrm{im}(\partial_{n+1})=B_n(X)</math>, so
| |
| | |
| :<math>H_n(X)=Z_n(X)/B_n(X). \, </math>
| |
| | |
| 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<sub>n</sub>(X)'' of a ''[[simplicial complex]]'' ''X'' are defined using the simplicial chain complex ''C(X)'', with ''C(X)<sub>n</sub>'' the free abelian group generated by the ''n''-simplices of ''X''. The ''[[singular homology]]'' groups ''H<sub>n</sub>(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 sequence|exact]] if the image of the (''n'' + 1)-th map is always equal to the kernel of the ''n''th 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<sup>n</sup>'' point in the direction of increasing ''n'' rather than decreasing ''n''; then the groups <math>\ker(d^n) = Z^n(X)</math> and <math>\mathrm{im}(d^{n - 1}) = B^n(X)</math> follow from the same description and
| |
| | |
| :<math>H^n(X) = Z^n(X)/B^n(X), \, </math>
| |
| | |
| as before.
| |
| | |
| Sometimes, [[reduced homology|reduced homology groups]] of a chain complex ''C(X)'' are defined as homologies of the augmented complex
| |
| :<math>\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{\epsilon}{\longrightarrow\,}
| |
| \Z {\longrightarrow\,}
| |
| 0</math>
| |
| where
| |
| :<math>\epsilon(\sum_i n_i \sigma_i)=\sum_i n_i</math>
| |
| for a combination ''Σ n<sub>i</sub>σ<sub>i</sub>'' of points σ<sub>i</sub> (fixed generators of ''C<sub>0</sub>''). The reduced homologies <math>\tilde{H}_i(X)</math> coincide with <math>H_i(X)</math> for ''i≠0''.
| |
| | |
| == Examples ==
| |
| The motivating example comes from [[algebraic topology]]: the '''[[simplicial homology]]''' of a [[simplicial complex]] ''X''. Here ''A<sub>n</sub>'' 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 [[vertex (geometry)|vertices]]
| |
| | |
| :<math> (a[0], a[1], \dots, a[n]) \, </math>
| |
| | |
| to the sum
| |
| | |
| :<math> \sum_{i=0}^n (-1)^i(a[0], \dots, a[i-1], a[i+1], \dots, a[n]) </math>
| |
| | |
| (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<sub>n</sub>'' to be the free abelian group (or free module) whose generators are all [[continuous function (topology)|continuous]] maps from ''n''-dimensional [[simplex|simplices]] into ''X''. The homomorphisms <math>\partial_n</math> arise from the boundary maps of simplices.
| |
| | |
| In [[abstract algebra]], one uses homology to define [[derived functor]]s, for example the [[Tor functor]]s. 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''<sub>1</sub> and a [[surjective]] homomorphism ''p''<sub>1</sub>: ''F''<sub>1</sub> → ''X''. Then one finds a free module ''F''<sub>2</sub> and a surjective homomorphism ''p''<sub>2</sub>: ''F''<sub>2</sub> → ker(''p''<sub>1</sub>). Continuing in this fashion, a sequence of free modules ''F<sub>n</sub>'' and homomorphisms ''p<sub>n</sub>'' can be defined. By applying the functor ''F'' to this sequence, one obtains a chain complex; the homology ''H<sub>n</sub>'' 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 theory|category]]: A morphism from the chain complex (''d<sub>n</sub>'': ''A<sub>n</sub>'' → ''A''<sub>''n''-1</sub>) to the chain complex (''e<sub>n</sub>'': ''B<sub>n</sub>'' → ''B''<sub>''n''-1</sub>) is a sequence of homomorphisms ''f<sub>n</sub>'': ''A<sub>n</sub>'' → ''B<sub>n</sub>'' such that <math>f_{n-1} \circ d_n = e_{n} \circ f_n </math> for all ''n''. The ''n''-th homology ''H<sub>n</sub>'' 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<sub>n</sub>'' are covariant [[functor]]s 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 ''H<sup>n</sup>'') form ''contravariant'' functors from the category that ''X'' belongs to into the category of abelian groups or modules.
| |
| | |
| == Properties ==
| |
| If (''d<sub>n</sub>'': ''A<sub>n</sub>'' → ''A''<sub>''n''-1</sub>) is a chain complex such that all but finitely many ''A<sub>n</sub>'' are zero, and the others are finitely generated abelian groups (or finite dimensional vector spaces), then we can define the ''[[Euler characteristic]]''
| |
| | |
| :<math> \chi = \sum (-1)^n \, \mathrm{rank}(A_n) </math>
| |
| | |
| (using the [[rank of an abelian group|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:
| |
| | |
| :<math> \chi = \sum (-1)^n \, \mathrm{rank}(H_n) </math>
| |
| | |
| 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]]
| |
| | |
| :<math> 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 </math>
| |
| | |
| of chain complexes gives rise to a [[long exact sequence]] of homology groups
| |
| | |
| :<math> \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. \,</math>
| |
| | |
| All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps ''H<sub>n</sub>(C)'' → ''H''<sub>''n''-1</sub>''(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]]<ref>{{Harvnb|Hilton|1988|p=284}}</ref><ref>For example [http://smf4.emath.fr/Publications/Gazette/2011/127/smf_gazette_127_15-44.pdf ''L'émergence de la notion de groupe d'homologie'', Nicolas Basbois (PDF)], in French, note 41, explicitly names Noether as inventing the [[homology group]].</ref> and, independently, by [[Leopold Vietoris]] and [[Walther Mayer]], in the period 1925–28.<ref>Hirzebruch, Friedrich, [http://www.mathe2.uni-bayreuth.de/axel/papers/hierzebruch:emmy_noether_and_topology.ps.gz Emmy Noether and Topology] in {{Harvnb|Teicher|1999|pp=61–63}}.</ref>
| |
| Prior to this, topological classes in [[combinatorial topology]] were not formally considered as [[abelian group]]s. The spread of homology groups marked the change of terminology and viewpoint from "combinatorial topology" to "algebraic topology".<ref>[http://math.vassar.edu/faculty/McCleary/BourbakiAlgTop.pdf ''Bourbaki and Algebraic Topology'' by John McCleary (PDF)] gives documentation (translated into English from French originals).</ref>
| |
| | |
| ==Applications==
| |
| Notable theorems proved using homology include the following:
| |
| * The [[Brouwer fixed point theorem]]: If ''f'' is any continuous map from the ball ''B<sup>n</sup>'' to itself, then there is a fixed point ''a'' ∈ ''B<sup>n</sup>'' with ''f''(''a'') = ''a''.
| |
| * [[Invariance of domain]]: If ''U'' is an [[open set|open subset]] of '''R'''<sup>''n''</sup> and ''f'' : ''U'' → '''R'''<sup>''n''</sup> is an [[injective]] [[continuous map]], then ''V'' = ''f''(''U'') is open and ''f'' is a [[homeomorphism]] between ''U'' and ''V''.
| |
| * The [[Hairy ball theorem]]: any vector field on the 2-sphere (or more generally, the 2''k''-sphere for any ''k'' ≥ 1) vanishes at some point.
| |
| * The [[Borsuk–Ulam theorem]]: any [[continuous function]] from an [[n-sphere|''n''-sphere]] into [[Euclidean space|Euclidean ''n''-space]] maps some pair of [[antipodal point]]s to the same point. (Two points on a sphere are called antipodal if they are in exactly opposite directions from the sphere's center.)
| |
| | |
| == See also ==
| |
| *[[Simplicial homology]]
| |
| *[[Singular homology]]
| |
| *[[Cellular homology]]
| |
| *[[Homological algebra]]
| |
| *[[Cohomology]]
| |
| | |
| == Notes ==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| * [[Henri Cartan|Cartan, Henri Paul]] and [[Samuel Eilenberg|Eilenberg, Samuel]] (1956) ''Homological Algebra'' Princeton University Press, Princeton, NJ, [http://worldcat.org/oclc/529171 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., [http://worldcat.org/oclc/1361982 OCLC 1361982]
| |
| * Hatcher, A., (2002) ''[http://www.math.cornell.edu/~hatcher/AT/ATchapters.html Algebraic Topology]'' Cambridge University Press, ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.
| |
| *[http://www.encyclopediaofmath.org/index.php/Homology_group ''Homology group'' at Encyclopaedia of Mathematics]
| |
| *{{citation|last=Hilton|first=Peter|year=1988|title=A Brief, Subjective History of Homology and Homotopy Theory in This Century |journal = Mathematics Magazine|volume=60|issue=5|pages=282–291| jstor = 2689545|publisher=Mathematical Association of America}}
| |
| *{{citation|title=The Heritage of Emmy Noether|first= M. (ed.)|last =Teicher|series=Israel Mathematical Conference Proceedings|publisher= [[Bar-Ilan University]]/[[American Mathematical Society]]/[[Oxford University Press]]|year= 1999|oclc= 223099225 |isbn= 978-0-19-851045-1}}
| |
| *{{planetmath reference|id=3720|title=Homology (Topological space)}}
| |
| | |
| [[Category:Homology theory|*]]
| |