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| In [[algebraic topology]], '''universal coefficient theorems''' establish relationships between
| | 56 year old Bed and Breakfast Operator Bud from Picton, usually spends time with interests which includes football, property developers in singapore and textiles. Lately took some time to take a trip to Gusuku Sites and Related [http://www.videome.org/profile/DeannaLDKi properties for sale in singapore] of the Kingdom of Ryukyu. |
| homology and cohomology theories. For instance, the ''integral [[homology theory]]'' of a [[topological space]] ''X'', and its ''homology with coefficients'' in any [[abelian group]] ''A'' are related as follows: the integral homology groups
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| :''H<sub>i</sub>''(''X'', '''Z''')
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| completely determine the groups
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| :''H<sub>i</sub>''(''X'', ''A'')
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| Here ''H<sub>i</sub>'' might be the [[simplicial homology]] or more general [[singular homology]] theory: the result itself is a pure piece of [[homological algebra]] about [[chain complex]]es of [[free abelian group]]s. The form of the result is that other coefficients ''A'' may be used, at the cost of using a [[Tor functor]].
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| For example it is common to take ''A'' to be '''Z'''/2'''Z''', so that coefficients are modulo 2. This becomes straightforward in the absence of 2-[[torsion_(algebra)|torsion]] in the homology. Quite generally, the result indicates the relationship that holds between the [[Betti number]]s ''b<sub>i</sub>'' of ''X'' and the Betti numbers ''b''<sub>''i'',''F''</sub> with coefficients in a [[field (mathematics)|field]] ''F''. These can differ, but only when the [[characteristic (algebra)|characteristic]] of ''F'' is a [[prime number]] ''p'' for which there is some ''p''-torsion in the homology.
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| ==Statement of the homology case ==
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| Consider the [[tensor product of modules]] ''H<sub>i</sub>''(''X'', '''Z''') ⊗ ''A''.The theorem states that there is an [[injective]] [[group homomorphism]] ι from this group to ''H<sub>i</sub>''(''X, A''), which has [[cokernel]] Tor(''H''<sub>''i''-1</sub>(''X'', '''Z'''), ''A''). In other words, there is a [[natural (category theory)|natural]] [[short exact sequence]]
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| :<math> 0 \rightarrow H_i(X, \mathbf{Z})\otimes A\rightarrow H_i(X,A)\rightarrow\mbox{Tor}(H_{i-1}(X, \mathbf{Z}),A)\rightarrow 0.</math>
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| Furthermore, this is a [[splitting lemma|split sequence]] (but the splitting is ''not'' natural).
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| The [[Tor functor|Tor group]] on the right can be thought of as the obstruction to ι being an isomorphism.
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| ==Universal coefficient theorem for cohomology==
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| Let ''G'' be a module over a principal ideal domain ''R'' (e.g.,<math>\mathbf{Z}</math> or a field.)
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| There is also a '''universal coefficient theorem for [[cohomology]]''' involving the [[Ext functor]], which asserts that there is a natural short exact sequence
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| :<math> 0 \rightarrow \operatorname{Ext}(\operatorname{H}_{i-1}(X; R), G) \rightarrow \operatorname{H}^i(X; G) \overset{h}\rightarrow \operatorname{Hom}_R(H_i(X; R), G)\rightarrow 0.</math> | |
| As in the homology case, the sequence splits, though not naturally.
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| In fact, suppose <math>\operatorname{H}_i(X;G) = \operatorname{ker}\operatorname{\partial}_i \otimes G / \operatorname{im}\operatorname{\partial}_{i+1} \otimes G</math> and <math>\operatorname{H}^*(X; G)</math> is defined as <math>\operatorname{ker}(\operatorname{Hom}(\partial, G)) / \operatorname{im}(\operatorname{Hom}(\partial, G))</math>. Then ''h'' above is the canonical map: <math>h([f])([x]) = f(x).</math> An alternative point-of-view can be based on representing cohomology via [[Eilenberg-MacLane_ space]] where the map ''h'' takes a homotopy class of maps from <math>X</math> to <math>K(G,i)</math> to the corresponding homomorphism induced in homology. Thus, the Eilenberg-MacLane space is a ''weak right [[adjoint]]'' to the homology [[functor]]. <ref>{{Harv|Kainen|1971}}</ref>
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| == Example: ''mod'' 2 cohomology of the real projective space==
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| Let ''X'' = '''P'''<sup>''n''</sup>('''R'''), the [[real projective space]]. We compute the singular cohomology of ''X'' with coefficients in ''R'' := '''Z'''<sub>2</sub>.
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| Knowing that the integer homology is given by:
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| :<math>H_i(X; \mathbf{Z}) =
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| \begin{cases}
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| \mathbf{Z} & i = 0 \mbox{ or } i = n \mbox{ odd,}\\
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| \mathbf{Z}/2\mathbf{Z} & 0<i<n,\ i\ \mbox{odd,}\\
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| 0 & \mbox{else.}
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| \end{cases}</math>
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| We have Ext(''R'', ''R'') = ''R'', Ext('''Z''', ''R'')= 0, so that the above exact sequences yield
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| :<math>\forall i = 0 \ldots n , \ H^i (X; R) = R</math>.
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| In fact the total [[cohomology ring]] structure is
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| :<math>H^*(X; R) = R [w] / \langle w^{n+1} \rangle </math>.
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| ==Corollaries==
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| A special case of the theorem is computing integral cohomology. For a finite CW complex ''X'', <math>H_i(X; \mathbf{Z})</math> is finitely generated, and so we have the following [[Fundamental_theorem_of_finitely_generated_abelian_groups#Classification|decomposition]].
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| :<math> H_i(X; \mathbf{Z}) \cong \mathbf{Z}^{\beta_i(X)}\oplus T_i .</math>
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| Where <math> \beta_i(X) </math> are the [[betti numbers]] of ''X''. One may check that
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| :<math> \mbox{Hom}(H_i(X),\mathbf{Z}) \cong \mbox{Hom}(\mathbf{Z}^{\beta_i(X)},\mathbf{Z}) \oplus \mbox{Hom}(T_i, \mathbf{Z}) \cong \mathbf{Z}^{\beta_i(X)} </math>, and <math> \mbox{Ext}(H_i(X),\mathbf{Z}) \cong \mbox{Ext}(\mathbf{Z}^{\beta_i(X)},\mathbf{Z}) \oplus \mbox{Ext}(T_i, \mathbf{Z}) \cong T_i. </math>
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| This gives the following statement for integral cohomology:
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| :<math> H^i(X;\mathbf{Z}) \cong \mathbf{Z}^{\beta_i(X)} \oplus T_{i-1}. </math>
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| For ''X'' an [[orientability|orientable]], [[closed manifold|closed]], and [[connected space|connected]] ''n''-[[manifold]], this corollary coupled with [[Poincaré duality]] gives that <math>\beta_i(X)=\beta_{n-i}(X) </math>.
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| {{reflist}}
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| ==References==
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| *[[Allen Hatcher]], ''Algebraic Topology'', Cambridge University Press, Cambridge, 2002. ISBN 0-521-79540-0. A modern, geometrically flavored introduction to algebraic topology. The book is available free in PDF and PostScript formats on the [http://www.math.cornell.edu/~hatcher/AT/ATpage.html author's homepage].
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| * {{cite journal
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| | last = Kainen
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| | first = P. C.
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| | authorlink = Paul Chester Kainen
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| | coauthors =
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| | title = Weak Adjoint Functors
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| | journal = Mathematische Zeitschrift
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| | volume = 122
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| | issue =
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| | pages = 1–9
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| | publisher =
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| | year = 1971
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| | pmid =
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| | pmc =
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| | doi =
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| }}
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| [[Category:Homological algebra]]
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| [[Category:Theorems in algebraic topology]]
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56 year old Bed and Breakfast Operator Bud from Picton, usually spends time with interests which includes football, property developers in singapore and textiles. Lately took some time to take a trip to Gusuku Sites and Related properties for sale in singapore of the Kingdom of Ryukyu.