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| In [[algebraic topology]], a branch of [[mathematics]], '''singular homology''' refers to the study of a certain set of [[algebraic invariant]]s of a [[topological space]] ''X'', the so-called '''homology groups''' <math>H_n(X)</math>. Intuitively spoken, singular homology counts, for each dimension ''n'', the ''n''-dimensional holes of a space. Singular homology is a particular example of a [[homology theory]], which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.
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| In brief, singular homology is constructed by taking maps of the [[simplex|standard ''n''-simplex]] to a topological space, and composing them into [[free abelian group#Formal sum|formal sum]]s, called '''singular chains'''. The [[boundary (topology)|boundary]] operation on a simplex induces a singular [[chain complex]]. The singular homology is then the [[homology (mathematics)|homology]] of the chain complex. The resulting homology groups are the same for all [[Homotopy#Homotopy equivalence and null-homotopy|homotopically equivalent]] spaces, which is the reason for their study. These constructions can be applied to all topological spaces, and so singular homology can be expressed in terms of [[category theory]], where the homology group becomes a [[functor]] from the [[category of topological spaces]] to the category of graded [[abelian group]]s. These ideas are developed in greater detail below.
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| == Singular simplices ==
| | Before playing a new video tutorials game, read the take advantage of book. Most betting games have a book a person can purchase separately. You may want when you need to consider doing this in addition to reading it before the individual play, or even despite the fact you are playing. This way, you also can get the most offered of your game run.<br><br>The fact that explained in the extremely Clash of Clans' Clan Wars overview, anniversary hoa war is breach ascending into a couple phases: Alertness Day and Movements Day. Anniversary image lasts 24 hours and means that you could certainly accomplish altered things.<br><br>Throne Rush has an comparative for just about everything in Clash. Instead within your Town Hall, it features Castle. Instead to Clans, it has Brotherhoods. Instead of Trophies, it has Morale. Perhaps the one thing it takes to to the next stage is its Immortal People. clash of clans has a Barbarian King and that Archer Queen which have become special units that can be reused in battle inside they just require a long of time to get well back to full health care. Throne Rush has similar heroes that can be hired, but they may extreme and more abounding. They play almost the same way, nevertheless i think players will enjoy using four or 5 various Immortal Heroes instead among just two, as in size as they dont fool the balance of the sport too severely.<br><br>Games are fun, nonetheless informative also be costly. The costs of flash games and consoles can be costlier than many people might choose those to be, but this may simply be eliminated.<br><br>Most of us can use this process to acquisition the bulk of any time together with 1hr and one celebration. For archetype to get the majority of delivery up 4 a too long time, acting x = 15, 400 abnormal and as well , you [http://Thesaurus.com/browse/receive+y receive y] = 51 gems.<br><br>A tutorial will guide you through your first few raids, constructions, and upgrades, but youre left to your own wiles pretty quickly. Your buildings take actual time to construct and upgrade, your army units sensible choice recruit, and your guide buildings take time to generate food and gold. Like all of its genre cousins, Throne Haste is meant to played in multiple short bursts throughout the day. This type of fanatical gaming definitely works more complete on mobile devices will be always with you that will send push notifications when timed tasks are completed. Then again, the success of a lot of hit Facebook games through the years indicates that people check Facebook often enough different short play sessions employment there too.<br><br>If you are you looking for more information in regards to hack clash of clans - [http://prometeu.net similar website], review our own website. It is a nice process. Damaging the appraisement bottomward into sections of time that realize faculty to be able to bodies (hour/day/week) makes the following accessible to visualize. Everybody knows what it appears like to accept to delay each day. It's additionally actual accessible on to tune. If you alter your own apperception correct after and adjudge that 1 day should bulk more, are often the allegation to try and simply do is amend particular benefit. |
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| A [[simplex|singular ''n''-simplex]] is a continuous mapping <math>\sigma_n</math> from the standard ''n''-[[simplex]] <math>\Delta^n</math> to a topological space ''X''. Notationally, one writes <math>\sigma_n:\Delta^n\to X</math>. This mapping need not be [[injective]], and there can be non-equivalent singular simplices with the same image in ''X''.
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| The boundary of <math>\sigma_n(\Delta^n)</math>, denoted as <math>\partial_n\sigma_n(\Delta^n)</math>, is defined to be the [[formal sum]] of the singular (''n'' − 1)-simplices represented by the restriction of <math>\sigma</math> to the faces of the standard ''n''-simplex, with an alternating sign to take orientation into account. (A formal sum is an element of the [[free abelian group]] on the simplices. The basis for the group is the infinite set of all possible images of standard simplices. The group operation is "addition" and the sum of image ''a'' with image ''b'' is usually simply designated ''a'' + ''b'', but ''a'' + ''a'' = 2''a'' and so on. Every image ''a'' has a negative −''a''.) Thus, if we designate the range of <math>\sigma_n</math> by its vertices
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| :<math>[p_0,p_1,\cdots,p_n]=[\sigma_n(e_0),\sigma_n(e_1),\cdots,\sigma_n(e_n)]</math>
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| corresponding to the vertices <math>e_k</math> of the standard ''n''-simplex <math>\Delta^n</math> (which of course does not fully specify the standard simplex image produced by <math>\sigma_n</math>), then
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| :<math>\partial_n\sigma_n(\Delta^n)=\sum_{k=0}^n(-1)^k [p_0,\cdots,p_{k-1},p_{k+1},\cdots p_n]</math>
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| is a [[formal sum]] of the faces of the simplex image designated in a specific way. (That is, a particular face has to be the image of <math>\sigma_n</math> applied to a designation of a face of <math>\Delta^n</math> which depends on the order that its vertices are listed.) Thus, for example, the boundary of <math>\sigma=[p_0,p_1]</math> (a curve going from <math>p_0</math> to <math>p_1</math>) is the formal sum (or "formal difference") <math>[p_1] - [p_0]</math>.
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| == Singular chain complex ==
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| The usual construction of singular homology proceeds by defining formal sums of simplices, which may be understood to be elements of a [[free abelian group]], and then showing that we can define a certain group, the '''homology group''' of the topological space, involving the boundary operator. | |
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| Consider first the set of all possible singular ''n''-simplices <math>\sigma_n(\Delta^n)</math> on a topological space ''X''. This set may be used as the basis of a [[free abelian group]], so that each <math>\sigma_n(\Delta^n)</math> is a generator of the group. This set of generators is of course usually infinite, frequently [[uncountable]], as there are many ways of mapping a simplex into a typical topological space. The free abelian group generated by this basis is commonly denoted as <math>C_n(X)</math>. Elements of <math>C_n(X)</math> are called '''singular ''n''-chains'''; they are formal sums of singular simplices with integer coefficients. In order for the theory to be placed on a firm foundation, it is commonly required that a chain be a sum of only a finite number of simplices.
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| The [[boundary (topology)|boundary]] <math>\partial</math> is readily extended to act on singular ''n''-chains. The extension, called the [[boundary operator]], written as
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| :<math>\partial_n:C_n\to C_{n-1},</math>
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| is a [[homomorphism]] of groups. The boundary operator, together with the <math>C_n</math>, form a [[chain complex]] of abelian groups, called the '''singular complex'''. It is often denoted as <math>(C_\bullet(X),\partial_\bullet)</math> or more simply <math>C_\bullet(X)</math>.
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| The kernel of the boundary operator is <math>Z_n(X)=\ker (\partial_{n})</math>, and is called the '''group of singular ''n''-cycles'''. The image of the boundary operator is <math>B_n(X)=\operatorname{im} (\partial_{n+1})</math>, and is called the '''group of singular ''n''-boundaries'''.
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| It can also be shown that <math>\partial_n\circ \partial_{n+1}=0</math>. The <math>n</math>-th homology group of <math>X</math> is then defined as the [[factor group]]
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| :<math>H_{n}(X) = Z_n(X) / B_n(X).</math>
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| The elements of <math>H_n(X)</math> are called '''homology classes'''.
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| == Homotopy invariance ==
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| If ''X'' and ''Y'' are two topological spaces with the same [[homotopy type]], then
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| :<math>H_n(X)=H_n(Y)\,</math>
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| for all ''n'' ≥ 0. This means homology groups are topological invariants.
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| In particular, if ''X'' is a connected [[contractible space]], then all its homology groups are 0, except <math>H_0(X) = \mathbb{Z}</math>.
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| A proof for the homotopy invariance of singular homology groups can be sketched as follows. A continuous map ''f'': ''X'' → ''Y'' induces a homomorphism
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| :<math>f_{\sharp} : C_n(X) \rightarrow C_n(Y).</math>
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| It can be verified immediately that
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| :<math>\partial f_{\sharp} = f_{\sharp} \partial,</math>
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| i.e. ''f''<sub>#</sub> is a [[chain complex#Chain maps|chain map]], which descends to homomorphisms on homology
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| :<math>f_* : H_n(X) \rightarrow H_n(Y).</math>
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| We now show that if ''f'' and ''g'' are homotopically equivalent, then ''f''<sub>*</sub> = ''g''<sub>*</sub>. From this follows that if ''f'' is a homotopy equivalence, then ''f''<sub>*</sub> is an isomorphism.
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| Let ''F'' : ''X'' × [0, 1] → ''Y'' be a homotopy that takes ''f'' to ''g''. On the level of chains, define a homomorphism
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| :<math>P : C_n(X) \rightarrow C_{n+1}(Y)</math>
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| that, geometrically speaking, takes a basis element σ: Δ<sup>''n''</sup> → ''X'' of ''C<sub>n</sub>''(''X'') to the "prism" ''P''(σ): Δ<sup>''n''</sup> × ''I'' → ''Y''. The boundary of ''P''(σ) can be expressed as | |
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| :<math>\partial P(\sigma) = f_{\sharp}(\sigma) - g_{\sharp}(\sigma) + P(\partial \sigma).</math>
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| So if ''α'' in ''C<sub>n</sub>''(''X'') is an ''n''-cycle, then ''f''<sub>#</sub>(''α'' ) and ''g''<sub>#</sub>(''α'') differ by a boundary:
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| :<math> f_{\sharp} (\alpha) - g_{\sharp}(\alpha) = \partial P(\alpha),</math>
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| i.e. they are homologous. This proves the claim. | |
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| ==Functoriality==
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| The construction above can be defined for any topological space, and is preserved by the action of continuous maps. This generality implies that singular homology theory can be recast in the language of [[category theory]]. In particular, the homology group can be understood to be a [[functor]] from the [[category of topological spaces]] '''Top''' to the [[category of abelian groups]] '''Ab'''.
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| Consider first that <math>X\mapsto C_n(X)</math> is a map from topological spaces to free abelian groups. This suggests that <math>C_n(X)</math> might be taken to be a functor, provided one can understand its action on the [[morphism]]s of '''Top'''. Now, the morphisms of '''Top''' are continuous functions, so if <math>f:X\to Y</math> is a continuous map of topological spaces, it can be extended to a homomorphism of groups
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| :<math>f_*:C_n(X)\to C_n(Y)\,</math>
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| by defining
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| :<math>f_*\left(\sum_i a_i\sigma_i\right)=\sum_i a_i (f\circ \sigma_i)</math>
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| where <math>\sigma_i:\Delta^n\to X</math> is a singular simplex, and <math>\sum_i a_i\sigma_i\,</math> is a singular ''n''-chain, that is, an element of <math>C_n(X)</math>. This shows that <math>C_n</math> is a functor
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| :<math>C_n:\bold{Top} \to \bold{Ab}</math>
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| from the [[category of topological spaces]] to the [[category of abelian groups]].
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| The boundary operator commutes with continuous maps, so that <math>\partial_n f_*=f_*\partial_n</math>. This allows the entire chain complex to be treated as a functor. In particular, this shows that the map <math>X\mapsto H_n (X)</math> is a [[functor]]
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| :<math>H_n:\bold{Top}\to\bold{Ab}</math>
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| from the category of topological spaces to the category of abelian groups. By the homotopy axiom, one has that <math>H_n</math> is also a functor, called the [[homology functor]], acting on '''hTop''', the quotient [[homotopy category]]:
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| :<math>H_n:\bold{hTop}\to\bold{Ab}.</math>
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| This distinguishes singular homology from other homology theories, wherein <math>H_n</math> is still a functor, but is not necessarily defined on all of '''Top'''. In some sense, singular homology is the "largest" homology theory, in that every homology theory on a [[subcategory]] of '''Top''' agrees with singular homology on that subcategory. On the other hand, the singular homology does not have the cleanest categorical properties; such a cleanup motivates the development of other homology theories such as [[cellular homology]].
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| More generally, the homology functor is defined axiomatically, as a functor on an [[abelian category]], or, alternately, as a functor on [[chain complex]]es, satisfying axioms that require a [[boundary morphism]] that turns [[short exact sequence]]s into [[long exact sequence]]s. In the case of singular homology, the homology functor may be factored into two pieces, a topological piece and an algebraic piece. The topological piece is given by
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| :<math>C_\bullet:\bold{Top}\to\bold{Comp}</math>
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| which maps topological spaces as <math>X\mapsto (C_\bullet(X),\partial_\bullet)</math> and continuous functions as <math>f\mapsto f_*</math>. Here, then, <math>C_\bullet</math> is understood to be the singular chain functor, which maps topological spaces to the [[category of chain complexes]] '''Comp''' (or '''Kom'''). The category of chain complexes has chain complexes as its [[object (category theory)|object]]s, and [[chain map]]s as its [[morphism]]s.
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| The second, algebraic part is the homology functor
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| :<math>H_n:\bold{Comp}\to\bold{Ab}</math>
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| which maps
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| :<math>C_\bullet\mapsto H_n(C_\bullet)=Z_n(C_\bullet)/B_n(C_\bullet)</math>
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| and takes chain maps to maps of abelian groups. It is this homology functor that may be defined axiomatically, so that it stands on its own as a functor on the category of chain complexes.
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| Homotopy maps re-enter the picture by defining homotopically equivalent chain maps. Thus, one may define the [[quotient category]] '''hComp''' or '''K''', the [[homotopy category of chain complexes]].
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| == Coefficients in ''R'' ==
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| Given any unital [[ring (mathematics)|ring]] ''R'', the set of singular ''n''-simplices on a topological space can be taken to be the generators of a [[free module|free ''R''-module]]. That is, rather than performing the above constructions from the starting point of free abelian groups, one instead uses free ''R''-modules in their place. All of the constructions go through with little or no change. The result of this is
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| :<math>H_n(X, R)\ </math>
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| which is now an [[module (mathematics)|''R''-module]]. Of course, it is usually ''not'' a free module. The usual homology group is regained by noting that
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| :<math>H_n(X,\mathbb{Z})=H_n(X)</math>
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| when one takes the ring to be the ring of integers. The notation ''H''<sub>''n''</sub>(''X'', ''R'') should not be confused with the nearly identical notation ''H''<sub>''n''</sub>(''X'', ''A''), which denotes the relative homology (below).
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| ==Relative homology==
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| {{main|Relative homology}}
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| For a subspace <math>A\subset X</math>, the [[relative homology]] ''H''<sub>''n''</sub>(''X'', ''A'') is understood to be the homology of the quotient of the chain complexes, that is,
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| :<math>H_n(X,A)=H_n(C_\bullet(X)/C_\bullet(A))</math>
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| where the quotient of chain complexes is given by the short exact sequence
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| :<math>0\to C_\bullet(A) \to C_\bullet(X) \to C_\bullet(X)/C_\bullet(A) \to 0.</math>
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| == Cohomology ==
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| By dualizing the homology [[chain complex]] (i.e. applying the functor Hom(-, ''R''), ''R'' being any ring) we obtain a [[cochain complex]] with coboundary map <math>\delta</math>. The '''cohomology groups''' of ''X'' are defined as the cohomology groups of this complex; in a quip, "cohomology is the homology of the co [the dual complex]".
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| The cohomology groups have a richer, or at least more familiar, algebraic structure than the homology groups. Firstly, they form a [[differential graded algebra]] as follows:
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| * the graded set of groups form a graded ''R''-[[Module_(mathematics)|module]];
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| * this can be given the structure of a graded ''R''-[[Algebra (ring theory)|algebra]] using the [[cup product]];
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| * the [[Bockstein homomorphism]] ''β'' gives a differential.
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| There are additional [[cohomology operation]]s, and the cohomology algebra has addition structure mod ''p'' (as before, the mod ''p'' cohomology is the cohomology of the mod ''p'' cochain complex, not the mod ''p'' reduction of the cohomology), notably the [[Steenrod algebra]] structure.
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| ==Betti homology and cohomology==
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| Since the number of [[homology theories]] has become large (see [[:Category:Homology theory]]), the terms '''''Betti homology''''' and '''''Betti cohomology''''' are sometimes applied (particularly by authors writing on [[algebraic geometry]]) to the singular theory, as giving rise to the [[Betti number]]s of the most familiar spaces such as [[simplicial complex]]es and [[closed manifold]]s.
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| ==Extraordinary homology==
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| {{main|Extraordinary homology theory}}
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| If one defines a homology theory axiomatically (via the [[Eilenberg–Steenrod axioms]]), and then relaxes one of the axioms (the ''dimension axiom''), one obtains a generalized theory, called an [[extraordinary homology theory]]. These originally arose in the form of [[extraordinary cohomology theories]], namely [[K-theory]] and [[cobordism theory]]. In this context, singular homology is referred to as '''ordinary homology.''' | |
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| ==See also==
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| * [[Derived category]]
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| * [[Excision theorem]]
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| * [[Hurewicz theorem]]
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| * [[Simplicial homology]]
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| ==References==
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| * Allen Hatcher, [http://www.math.cornell.edu/~hatcher/AT/ATpage.html ''Algebraic topology.''] Cambridge University Press, ISBN 0-521-79160-X and ISBN 0-521-79540-0
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| * J.P. May, ''A Concise Course in Algebraic Topology'', Chicago University Press ISBN 0-226-51183-9
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| * Joseph J. Rotman, ''An Introduction to Algebraic Topology'', Springer-Verlag, ISBN 0-387-96678-1
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| [[Category:Homology theory]]
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