Dual universal coefficient theorem - Revision history

Vipul: /* Related facts */

2015-05-09T22:44:56Z

Related facts

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	==Related facts==		==Related facts==

	* [[Universal ~~coefficients~~ theorem for homology]]		* [[Universal coefficient theorem for homology]]
	* [[Universal ~~coefficients~~ theorem for cohomology]]		* [[Universal coefficient theorem for cohomology]]
	* [[Kunneth formula for homology]]		* [[Kunneth formula for homology]]
	* [[Kunneth formula for cohomology]]		* [[Kunneth formula for cohomology]]

Vipul at 22:44, 9 May 2015

2015-05-09T22:44:33Z

← Older revision		Revision as of 22:44, 9 May 2015
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	{{quotation\|A more detailed page on the same theorem, but from a purely algebraic perspective, is at [[Groupprops:Dual universal coefficient theorem]]}}		{{quotation\|A more detailed page on the same theorem, but from a purely algebraic perspective, is at [[Groupprops:Dual universal coefficient theorem for group cohomology]]}}

	==Statement==		==Statement==

Vipul at 22:41, 9 May 2015

2015-05-09T22:41:38Z

← Older revision		Revision as of 22:41, 9 May 2015
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			{{quotation\|A more detailed page on the same theorem, but from a purely algebraic perspective, is at [[Groupprops:Dual universal coefficient theorem]]}}

	==Statement==		==Statement==

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	In particular, if <math>H_1(X;\mathbb{Z})</math> is finitely generated, then <math>H^1(X;\mathbb{Z})</matH> is free abelian and equals the torsion-free part of <math>H_1(X;\mathbb{Z})</math>.		In particular, if <math>H_1(X;\mathbb{Z})</math> is finitely generated, then <math>H^1(X;\mathbb{Z})</matH> is free abelian and equals the torsion-free part of <math>H_1(X;\mathbb{Z})</math>.


	In the case that both <math>H_{n-1}(X;\mathbb{Z})</math> and <math>H_n(X;\mathbb{Z})</math> are free abelian groups, and the latter has finite rank, we get:		In the case that both <math>H_{n-1}(X;\mathbb{Z})</math> and <math>H_n(X;\mathbb{Z})</math> are free abelian groups, and the latter has finite rank, we get:

	<math>H^n(X;\mathbb{Z}) \cong H_n(X;\mathbb{Z})</math>		<math>H^n(X;\mathbb{Z}) \cong H_n(X;\mathbb{Z})</math>

	~~In particular, if all the ho~~

Vipul: Vipul moved page Dual universal coefficients theorem to Dual universal coefficient theorem

2015-05-09T22:40:45Z

Vipul moved page Dual universal coefficients theorem to Dual universal coefficient theorem

← Older revision	Revision as of 22:40, 9 May 2015
(No difference)

Vipul: Created page with "==Statement== ===For coefficients in an abelian group=== Suppose $X$ is a topological space and $M$ is an abelian group. The '''dual universal coe..."

2011-07-27T16:28:33Z

Created page with "==Statement== ===For coefficients in an abelian group=== Suppose <math>X</math> is a topological space and <math>M</math> is an abelian group. The '''dual universal coe..."

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==Statement==

===For coefficients in an abelian group===

Suppose <math>X</math> is a [[topological space]] and <math>M</math> is an [[abelian group]]. The '''dual universal coefficients theorem''' relates the [[homology group]]s of <math>X</math> with coefficients in <math>\mathbb{Z}</math> and the [[cohomology group]]s of <math>X</math> with coefficients in <math>M</math> as follows:

First, for any <math>n \ge 0</math>, there is a natural [[short exact sequence of abelian groups]]:

<math>0 \to \operatorname{Ext}(H_{n-1}(X;\mathbb{Z}),M) \to H^n(X;M) \to \operatorname{Hom}(H_n(X;\mathbb{Z}),M) \to 0</math>

Second, the sequence splits (not necessarily naturally), and we get:

<math>H^n(X;M) \cong \operatorname{Hom}(H_n(X;\mathbb{Z}),M) \oplus \operatorname{Ext}(H_{n-1}(X;\mathbb{Z}),M)</math>

===For coefficients in the integers===

This is the special case where <math>M = \mathbb{Z}</math>. In this case, we case:

<math>H^n(X;\mathbb{Z}) \cong \operatorname{Hom}(H_n(X;\mathbb{Z}),\mathbb{Z}) \oplus \operatorname{Ext}(H_{n-1}(X;\mathbb{Z}),\mathbb{Z})</math>

==Related facts==

* [[Universal coefficients theorem for homology]]
* [[Universal coefficients theorem for cohomology]]
* [[Kunneth formula for homology]]
* [[Kunneth formula for cohomology]]

==Particular cases==

===Case of free abelian groups===

In the case that <math>H_{n-1}(X;\mathbb{Z})</math> is a [[free abelian group]], we get:

<math>H^n(X;\mathbb{Z}) \cong \operatorname{Hom}(H_n(X;\mathbb{Z}),\mathbb{Z})</math>

Further, if <math>H_n(X;\mathbb{Z})</math> is finitely generated, then, under these circumstances, <math>H^n(X;\mathbb{Z})</math> is simply the torsion-free part of <math>H_n(X;\mathbb{Z})</math>.

Note that this always applies to the case <math>n = 1</math>, because <math>H_0</math> is a free abelian group of rank equal to the number of connected components. Thus, we get:

<math>H^1(X;\mathbb{Z}) \cong \operatorname{Hom}(H_1(X;\mathbb{Z}),\mathbb{Z})</math>

In particular, if <math>H_1(X;\mathbb{Z})</math> is finitely generated, then <math>H^1(X;\mathbb{Z})</matH> is free abelian and equals the torsion-free part of <math>H_1(X;\mathbb{Z})</math>.

In the case that both <math>H_{n-1}(X;\mathbb{Z})</math> and <math>H_n(X;\mathbb{Z})</math> are free abelian groups, and the latter has finite rank, we get:

<math>H^n(X;\mathbb{Z}) \cong H_n(X;\mathbb{Z})</math>

In particular, if all the ho

Dual universal coefficient theorem - Revision history

Vipul: /* Related facts */

Vipul at 22:44, 9 May 2015

Vipul at 22:41, 9 May 2015

Vipul: Vipul moved page Dual universal coefficients theorem to Dual universal coefficient theorem

Vipul: Created page with "==Statement== ===For coefficients in an abelian group=== Suppose X is a topological space and M is an abelian group. The '''dual universal coe..."

Vipul: Created page with "==Statement== ===For coefficients in an abelian group=== Suppose $X$ is a topological space and $M$ is an abelian group. The '''dual universal coe..."