Kunneth formula for homology - Revision history

Vipul: /* Related facts */

2011-07-27T15:32:57Z

Related facts

← Older revision		Revision as of 15:32, 27 July 2011
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	* [[Universal coefficients theorem for homology]]		* [[Universal coefficients theorem for homology]]
	* [[Universal coefficients theorem for cohomology]]		* [[Universal coefficients theorem for cohomology]]
			* [[Dual universal coefficients theorem]] (computes cohomology in terms of homology)

	==Particular cases==		==Particular cases==

Vipul at 15:31, 27 July 2011

2011-07-27T15:31:43Z

← Older revision		Revision as of 15:31, 27 July 2011
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	Here, <math>\operatorname{Tor}</math> is torsion of modules over the ring <math>R</matH>.		Here, <math>\operatorname{Tor}</math> is torsion of modules over the ring <math>R</matH>.

			==Related facts==

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

	==Particular cases==		==Particular cases==

Vipul: moved Kunneth formula to Kunneth formula for homology

2011-07-27T15:30:58Z

moved Kunneth formula to Kunneth formula for homology

← Older revision	Revision as of 15:30, 27 July 2011
(No difference)

Vipul: /* Statement */

2011-07-26T03:23:36Z

Statement

← Older revision		Revision as of 03:23, 26 July 2011
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	Suppose <math>X</math> and <math>Y</math> are [[topological space]]s. We then have the following relation for the [[homology group]]s of <math>X</math>, <math>Y</math>, and the [[product topology\|product space]] <math>X \times Y</math>.		Suppose <math>X</math> and <math>Y</math> are [[topological space]]s. We then have the following relation for the [[homology group]]s of <math>X</math>, <math>Y</math>, and the [[product topology\|product space]] <math>X \times Y</math>.

	For any <math>n \ge 0</math> and any module <math>M</math> over a ~~commutative unital ring~~ <math>R</math> for coefficients, we have:		For any <math>n \ge 0</math> and any module <math>M</math> over a [[principal ideal domain]] <math>R</math> for coefficients, we have:

	<math>H_n(X \times Y;M) \cong \left(\sum_{i + j = n} H_i(X;M) \otimes H_j(Y;M)\right) \oplus \left(\sum_{p + q = n-1} \operatorname{Tor}(H_p(X;M),H_q(Y;M))\right)</math>		<math>H_n(X \times Y;M) \cong \left(\sum_{i + j = n} H_i(X;M) \otimes H_j(Y;M)\right) \oplus \left(\sum_{p + q = n-1} \operatorname{Tor}(H_p(X;M),H_q(Y;M))\right)</math>

Vipul: /* Statement */

2011-07-21T15:39:01Z

Statement

← Older revision		Revision as of 15:39, 21 July 2011
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	==Statement==		==Statement==

	Suppose <math>X</math> and <math>Y</math> are [[topological space]]s. We then have the following relation for the [[homology group]]s of <math>X<math>, <math>Y</math>, and the [[product topology\|product space]] <math>X \times Y</math>.		Suppose <math>X</math> and <math>Y</math> are [[topological space]]s. We then have the following relation for the [[homology group]]s of <math>X</math>, <math>Y</math>, and the [[product topology\|product space]] <math>X \times Y</math>.

	For any <math>n \ge 0</math> and any module <math>M</math> over a commutative unital ring <math>R</math> for coefficients, we have:		For any <math>n \ge 0</math> and any module <math>M</math> over a commutative unital ring <math>R</math> for coefficients, we have:

Vipul at 23:51, 1 April 2011

2011-04-01T23:51:12Z

@@ Line 1: / Line 1: @@
 ==Statement==
-Suppose <math>X</math> and <math>Y</math> are [[topological space]]s. We then have, for any <math>n \ge 0</math>:
+Suppose <math>X</math> and <math>Y</math> are [[topological space]]s. We then have the following relation for the [[homology group]]s of <math>X<math>, <math>Y</math>, and the [[product topology|product space]] <math>X \times Y</math>.
-<math>H_n(X \times Y) \cong \sum_{i + j = n} H_i(X) \otimes H_j(Y) \oplus \sum_{p + q = n-1} Tor(H_p(X),H_q(Y))</math>
+For any <math>n \ge 0</math> and any module <math>M</math> over a commutative unital ring <math>R</math> for coefficients, we have:
-==Results used==
+<math>H_n(X \times Y;M) \cong \left(\sum_{i + j = n} H_i(X;M) \otimes H_j(Y;M)\right) \oplus \left(\sum_{p + q = n-1} \operatorname{Tor}(H_p(X;M),H_q(Y;M))\right)</math>
 The Kunneth formula combines the [[Kunneth theorem]] and the [[Eilenberg-Zilber theorem]].

Vipul: 2 revisions

2008-05-11T19:48:15Z

2 revisions

← Older revision	Revision as of 19:48, 11 May 2008
(No difference)

Vipul at 20:17, 3 November 2007

2007-11-03T20:17:53Z

← Older revision		Revision as of 20:17, 3 November 2007
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	Suppose <math>X</math> and <math>Y</math> are [[topological space]]s. We then have, for any <math>n \ge 0</math>:		Suppose <math>X</math> and <math>Y</math> are [[topological space]]s. We then have, for any <math>n \ge 0</math>:

	<math>H_n(X \times Y) \cong \sum_{i + j = n} H_i(X) \otimes H_j(Y) \oplus \sum_{p + q = n-1} ~~\text{~~Tor}(H_p(X),H_q(Y))</math>		<math>H_n(X \times Y) \cong \sum_{i + j = n} H_i(X) \otimes H_j(Y) \oplus \sum_{p + q = n-1} Tor(H_p(X),H_q(Y))</math>

	==Results used==		==Results used==

Vipul at 20:17, 3 November 2007

2007-11-03T20:17:25Z

New page

==Statement==

Suppose <math>X</math> and <math>Y</math> are [[topological space]]s. We then have, for any <math>n \ge 0</math>:

<math>H_n(X \times Y) \cong \sum_{i + j = n} H_i(X) \otimes H_j(Y) \oplus \sum_{p + q = n-1} \text{Tor}(H_p(X),H_q(Y))</math>

==Results used==

The Kunneth formula combines the [[Kunneth theorem]] and the [[Eilenberg-Zilber theorem]].

==Proof==

{{fillin}}