Homology of spheres - Revision history

Vipul: /* Related invariants */

2011-01-09T22:51:59Z

Related invariants

← Older revision		Revision as of 22:51, 9 January 2011
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	! Invariant !! General description !! Description of value for spheres		! Invariant !! General description !! Description of value for spheres
	\|-		\|-
	\| [[Betti number]]s \|\| The <math>n^{th}</math> Betti number <math>b_k</math> is the rank of the <math>k^{th}</math> homology group. \|\| For <math>n = 0</math>, <math>b_0 = 2</math>, all other <math>b_k</math> are <math>0</math>; for <math>n > 0</math>, <math>b_0 = b_n = 1</math>, all other <math>b_k</math>s are <math>0</math>.		\| [[Betti number]]s \|\| The <math>k^{th}</math> Betti number <math>b_k</math> is the rank of the <math>k^{th}</math> homology group. \|\| For <math>n = 0</math>, <math>b_0 = 2</math>, all other <math>b_k</math> are <math>0</math>; for <math>n > 0</math>, <math>b_0 = b_n = 1</math>, all other <math>b_k</math>s are <math>0</math>.
	\|-		\|-
	\| [[Poincare polynomial]] \|\| Generating polynomial for Betti numbers \|\| <math>2</math> for <math>n = 0</math>, <math>1 + x^n</math> for <math>n > 0</math>		\| [[Poincare polynomial]] \|\| Generating polynomial for Betti numbers \|\| <math>2</math> for <math>n = 0</math>, <math>1 + x^n</math> for <math>n > 0</math>

Vipul: /* Related invariants */

2010-12-31T18:46:32Z

Related invariants

← Older revision		Revision as of 18:46, 31 December 2010
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	\|-		\|-
	\| [[Betti number]]s \|\| The <math>n^{th}</math> Betti number <math>b_k</math> is the rank of the <math>k^{th}</math> homology group. \|\| For <math>n = 0</math>, <math>b_0 = 2</math>, all other <math>b_k</math> are <math>0</math>; for <math>n > 0</math>, <math>b_0 = b_n = 1</math>, all other <math>b_k</math>s are <math>0</math>.		\| [[Betti number]]s \|\| The <math>n^{th}</math> Betti number <math>b_k</math> is the rank of the <math>k^{th}</math> homology group. \|\| For <math>n = 0</math>, <math>b_0 = 2</math>, all other <math>b_k</math> are <math>0</math>; for <math>n > 0</math>, <math>b_0 = b_n = 1</math>, all other <math>b_k</math>s are <math>0</math>.
			\|-
			\| [[Poincare polynomial]] \|\| Generating polynomial for Betti numbers \|\| <math>2</math> for <math>n = 0</math>, <math>1 + x^n</math> for <math>n > 0</math>
	\|-		\|-
	\| [[Euler characteristic]] \|\| <math>\sum_{k=0}^\infty (-1)^k b_k</math> \|\| <math>2</math> for <matH>n</math> even, <math>0</math> for <math>n</math> odd.		\| [[Euler characteristic]] \|\| <math>\sum_{k=0}^\infty (-1)^k b_k</math> \|\| <math>2</math> for <matH>n</math> even, <math>0</math> for <math>n</math> odd.
	\|}		\|}

	==Facts used==		==Facts used==

Vipul at 18:15, 31 December 2010

2010-12-31T18:15:08Z

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	* <math>n > 0</math> case: <math>H_0(S^n) \cong H_n(S^n) \cong M</math> and <math>H_k(S^n)</math> is trivial for <math>k \ne 0,n</math>.		* <math>n > 0</math> case: <math>H_0(S^n) \cong H_n(S^n) \cong M</math> and <math>H_k(S^n)</math> is trivial for <math>k \ne 0,n</math>.

			==Related invariants==

			These are all invariants that can be computed in terms of the homology groups.

			{\| class="sortable" border="1"
			! Invariant !! General description !! Description of value for spheres
			\|-
			\| [[Betti number]]s \|\| The <math>n^{th}</math> Betti number <math>b_k</math> is the rank of the <math>k^{th}</math> homology group. \|\| For <math>n = 0</math>, <math>b_0 = 2</math>, all other <math>b_k</math> are <math>0</math>; for <math>n > 0</math>, <math>b_0 = b_n = 1</math>, all other <math>b_k</math>s are <math>0</math>.
			\|-
			\| [[Euler characteristic]] \|\| <math>\sum_{k=0}^\infty (-1)^k b_k</math> \|\| <math>2</math> for <matH>n</math> even, <math>0</math> for <math>n</math> odd.
			\|}
	==Facts used==		==Facts used==

Vipul at 18:06, 31 December 2010

2010-12-31T18:06:37Z

← Older revision		Revision as of 18:06, 31 December 2010
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			{{homotopy invariant computation\|
			invariant = homology group\|
			space = sphere}}

	In this article, we briefly describe the [[computation of::homology group]]s of [[specific information about::sphere]]s, a proof using the [[Mayer-Vietoris homology sequence]], and explanations in terms of cellular and simplicial homology.		In this article, we briefly describe the [[computation of::homology group]]s of [[specific information about::sphere]]s, a proof using the [[Mayer-Vietoris homology sequence]], and explanations in terms of cellular and simplicial homology.

Vipul at 17:39, 31 December 2010

2010-12-31T17:39:08Z

← Older revision		Revision as of 17:39, 31 December 2010
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	In this article, we briefly describe ~~how to compute~~ the homology ~~groups~~ of ~~spheres~~ using the [[Mayer-Vietoris homology sequence]].		In this article, we briefly describe the [[computation of::homology group]]s of [[specific information about::sphere]]s, a proof using the [[Mayer-Vietoris homology sequence]], and explanations in terms of cellular and simplicial homology.

	==Statement==		==Statement==
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	# [[uses::Homology for suspension]]		# [[uses::Homology for suspension]]
			# [[uses::CW structure of spheres]]
	==Proof==		# [[uses::Simplicial structure of spheres]]
			==Proof using singular homology==

	===Equivalence of reduced and unreduced version===		===Equivalence of reduced and unreduced version===
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	In general, we use induction, starting with the base case <math>n = 0</math>. The inductive step follows from fact (1) and the fact that each <math>S^n</math> is the suspension of <math>S^{n-1}</math>.		In general, we use induction, starting with the base case <math>n = 0</math>. The inductive step follows from fact (1) and the fact that each <math>S^n</math> is the suspension of <math>S^{n-1}</math>.

			==Proof using cellular homology==

			See (2): [[CW structure of spheres]].

			==Proof using simplicial homology==

			See (3): [[Simplicial structure of spheres]].

Vipul: moved Homology computation for spheres to Homology of spheres

2010-12-31T04:15:04Z

moved Homology computation for spheres to Homology of spheres

← Older revision	Revision as of 04:15, 31 December 2010
(No difference)

Vipul at 03:43, 24 December 2010

2010-12-24T03:43:30Z

← Older revision		Revision as of 03:43, 24 December 2010
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	In this article, we briefly describe how to compute the homology groups of spheres using the [[Mayer-Vietoris homology sequence]].		In this article, we briefly describe how to compute the homology groups of spheres using the [[Mayer-Vietoris homology sequence]].

	{{~~fillin~~}}		==Statement==

			===Reduced version over integers===

			For <math>n</math> a nonnegative integer, we have the following result for the [[reduced homology group]]s:

			<math>\tilde{H}_k(S^n) = 0, k \ne n</math>:

			and:

			<math>\tilde{H}_n(S^n) \cong \mathbb{Z}</math>

			===Unreduced version over integers===

			We need to make cases based on whether <math>n = 0</math> or <math>n</math> is a positive integer:

			* <math>n = 0</math> case: <math>H_0(S^0) \cong \mathbb{Z} \oplus \mathbb{Z}</math> and <math>H_k(S^0)</math> is trivial for <math>k > 0</math>.
			* <math>n > 0</math> case: <math>H_0(S^n) \cong H_n(S^n) \cong \mathbb{Z}</math> and <math>H_k(S^n)</math> is trivial for <math>k \ne 0,n</math>.

			===Reduced version over a module <math>M</math> over a ring <math>R</math>===

			For <math>n</math> a nonnegative integer, we have the following result for the [[reduced homology group]]s:

			<math>\tilde{H}_k(S^n) = 0, k \ne n</math>:

			and:

			<math>\tilde{H}_n(S^n) \cong M</math>

			===Unreduced version over a module <math>M</math> over a ring <math>R</math>===

			We need to make cases based on whether <math>n = 0</math> or <math>n</math> is a positive integer:

			* <math>n = 0</math> case: <math>H_0(S^0) \cong M \oplus M</math> and <math>H_k(S^0)</math> is trivial for <math>k > 0</math>.
			* <math>n > 0</math> case: <math>H_0(S^n) \cong H_n(S^n) \cong M</math> and <math>H_k(S^n)</math> is trivial for <math>k \ne 0,n</math>.

			==Facts used==

			# [[uses::Homology for suspension]]

			==Proof==

			===Equivalence of reduced and unreduced version===

			The equivalence follows from the fact that reduced and unreduced homology groups coincide for <math>k > 0</math> and for <math>k = 0</math>, the unreduced homology group is obtained from the reduced one by adding a copy of <math>\mathbb{Z}</math> (or, if working over another ring or module, the base ring or module).

			===Proof of reduced version===

			The case <math>n = 0</math> is clear: the space <math>S^0</math> is a discrete two-point space, hence it has two single-point path components, so the zeroth homology group is <math>M^{2 - 1} = M^1 = M</math>. Higher homology groups are trivial because the cycle and boundary groups both coincide with the group of ''all'' functions to <math>S^0</math>, so the homology group is trivial.

			In general, we use induction, starting with the base case <math>n = 0</math>. The inductive step follows from fact (1) and the fact that each <math>S^n</math> is the suspension of <math>S^{n-1}</math>.

Vipul: Created page with 'In this article, we briefly describe how to compute the homology groups of spheres using the Mayer-Vietoris homology sequence. {{fillin}}'

2010-12-20T01:18:11Z

Created page with 'In this article, we briefly describe how to compute the homology groups of spheres using the Mayer-Vietoris homology sequence. {{fillin}}'

New page

In this article, we briefly describe how to compute the homology groups of spheres using the [[Mayer-Vietoris homology sequence]].

{{fillin}}