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In [[algebraic topology]], '''universal coefficient theorems'''  establish relationships between
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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
 
:''H<sub>i</sub>''(''X'', '''Z''')
 
completely determine the groups
 
:''H<sub>i</sub>''(''X'', ''A'')
 
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]].
 
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.
 
==Statement of the homology case ==
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]]
 
:<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>
 
Furthermore, this is a [[splitting lemma|split sequence]] (but the splitting is ''not'' natural).
 
The [[Tor functor|Tor group]] on the right can be thought of as the obstruction to ι being an isomorphism.
 
==Universal coefficient theorem for cohomology==
Let ''G'' be a module over a principal ideal domain ''R'' (e.g.,<math>\mathbf{Z}</math> or a field.)
 
There is also a '''universal coefficient theorem for [[cohomology]]''' involving the [[Ext functor]], which asserts that there is a natural short exact sequence
:<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.
 
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>
 
== Example: ''mod'' 2 cohomology of the real projective space==
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>.
 
Knowing that the integer homology is given by:
 
:<math>H_i(X; \mathbf{Z}) =
\begin{cases}
\mathbf{Z} & i = 0 \mbox{ or } i = n \mbox{ odd,}\\
\mathbf{Z}/2\mathbf{Z} & 0<i<n,\ i\ \mbox{odd,}\\
0 & \mbox{else.}
\end{cases}</math>
 
We have Ext(''R'', ''R'') = ''R'', Ext('''Z''', ''R'')= 0, so that the above exact sequences yield
:<math>\forall i = 0 \ldots n , \ H^i (X; R) = R</math>.
 
In fact the total [[cohomology ring]] structure is
:<math>H^*(X; R) = R [w] / \langle w^{n+1} \rangle </math>.
 
==Corollaries==
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]].
 
:<math> H_i(X; \mathbf{Z}) \cong \mathbf{Z}^{\beta_i(X)}\oplus T_i .</math>
 
Where <math> \beta_i(X) </math> are the [[betti numbers]] of ''X''. One may check that
 
:<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>
 
This gives the following statement for integral cohomology:
 
:<math> H^i(X;\mathbf{Z}) \cong \mathbf{Z}^{\beta_i(X)} \oplus T_{i-1}. </math>
 
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>.
 
{{reflist}}
 
==References==
*[[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].
 
* {{cite journal
  | last = Kainen
  | first = P. C.
  | authorlink = Paul Chester Kainen
  | coauthors =
  | title = Weak Adjoint Functors
  | journal = Mathematische Zeitschrift
  | volume = 122
  | issue =
  | pages = 1–9
  | publisher =
  | year = 1971
  | pmid = 
  | pmc =
  | doi =
  }}
 
[[Category:Homological algebra]]
[[Category:Theorems in algebraic topology]]

Revision as of 20:48, 9 February 2014

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.