2-sphere - Revision history

Vipul: /* Manifold-type invariants */

2011-07-24T16:27:34Z

Manifold-type invariants

← Older revision		Revision as of 16:27, 24 July 2011
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	==Manifold-type invariants==		==Manifold-type invariants==

	~~===Dimension invariants===~~

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
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	\|-		\|-
	\| smallest dimension of Euclidean space in which it can be embedded as a flat submanifold \|\| -- \|\| It's not possible to embed this as a flat manifold anywhere, because the sphere is intrinsically curved (something to do with Gauss-Bonnet theorem, deeper stuff, Euler characteristic being nonzero, links need to be added)		\| smallest dimension of Euclidean space in which it can be embedded as a flat submanifold \|\| -- \|\| It's not possible to embed this as a flat manifold anywhere, because the sphere is intrinsically curved (something to do with Gauss-Bonnet theorem, deeper stuff, Euler characteristic being nonzero, links need to be added)
			\|-
			\| minimum number of charts needed in an atlas for this manifold \|\| 2 \|\| the complement of any single point gives a chart mapping to all of <math>\R^2</math>, thus, we can construct an atlas by using the complements of two distinct points.
	\|}		\|}

Vipul: /* Algebraic and coalgebraic structure */

2011-07-24T16:23:21Z

Algebraic and coalgebraic structure

← Older revision		Revision as of 16:23, 24 July 2011
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	where <math>\vee</math> denotes the wedge sum and the map is a [[continuous based map]], i.e., a continuous map preserving basepoint. This map is cocommutative and coassociative up to homotopy, and it is used to give an abelian group structure to the set of homotopy classes from the based 2-sphere to any [[based topological space]]. This group is termed the [[second homotopy group]].		where <math>\vee</math> denotes the wedge sum and the map is a [[continuous based map]], i.e., a continuous map preserving basepoint. This map is cocommutative and coassociative up to homotopy, and it is used to give an abelian group structure to the set of homotopy classes from the based 2-sphere to any [[based topological space]]. This group is termed the [[second homotopy group]].

			==Manifold-type invariants==

			===Dimension invariants===

			{\| class="sortable" border="1"
			! Invariant !! Value !! Explanation
			\|-
			\| [[dimension]] (we can use any of the dimension definitions since this is a connected manifold) \|\| 2 \|\|
			\|-
			\| smallest dimension of Euclidean space in which it can be immersed \|\| 3 \|\| The usual embedding in <math>\R^3</math> as a sphere with a center and radius
			\|-
			\| smallest dimension of Euclidean space in which it can be embedded \|\| 3 \|\| The usual embedding in <math>\R^3</math> as a sphere with a center and radius
			\|-
			\| smallest dimension of Euclidean space in which it can be embedded as a flat submanifold \|\| -- \|\| It's not possible to embed this as a flat manifold anywhere, because the sphere is intrinsically curved (something to do with Gauss-Bonnet theorem, deeper stuff, Euler characteristic being nonzero, links need to be added)
			\|}

Vipul: /* Equivalent spaces */

2011-07-23T23:59:26Z

Equivalent spaces

← Older revision		Revision as of 23:59, 23 July 2011
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	\|-		\|-
	\| hollow cube in <math>\R^3</math> \|\| homeomorphism arising from a straight line homotopy		\| hollow cube in <math>\R^3</math> \|\| homeomorphism arising from a straight line homotopy
			\|-
			\| quotient of closed unit disk in <math>\R^2</math> by the identification of all points in its boundary with each other, i.e., <math>D^2/\partial D^2</math> \|\| via identification with one-point compactification of <math>\R^2</math>: the interior of the disk can be identified with <math>\R^2</math>, and the boundary point is identified with the point at infinity.
	\|-		\|-
	\| [[suspension]] of [[circle]] \|\| (easy, fill in)		\| [[suspension]] of [[circle]] \|\| (easy, fill in)

Vipul at 00:36, 22 July 2011

2011-07-22T00:36:02Z

← Older revision		Revision as of 00:36, 22 July 2011
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	{{particular topospace}}		{{particular topospace}}
			[[dimension::2\| ]]
	==Definition==		==Definition==

Vipul: /* Homotopy groups */

2011-07-21T23:41:17Z

Homotopy groups

← Older revision		Revision as of 23:41, 21 July 2011
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	===Homotopy groups===		===Homotopy groups===

	{{further\|[[homotopy ~~computation for~~ spheres]]}}		{{further\|[[homotopy of spheres]]}}

	{\| class="sortable" border="1"		{\| class="sortable" border="1"

Vipul: /* Homology groups */

2011-07-21T23:40:35Z

Homology groups

← Older revision		Revision as of 23:40, 21 July 2011
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	===Homology groups===		===Homology groups===

	{{further\|[[homology ~~computation for~~ spheres]]}}		{{further\|[[homology of spheres]]}}

	The homology groups with coefficients in <math>\mathbb{Z}</math> are as follows: <math>H_0(S^2) \cong H_2(S^2) \cong \mathbb{Z}</math>, and all other homology groups are zero. The reduced homology groups with coefficients in <math>\mathbb{Z}</math> are as follows: <math>\tilde{H}_2(S^2) \cong \mathbb{Z}</math>, and all other reduced homology groups are zero.		The homology groups with coefficients in <math>\mathbb{Z}</math> are as follows: <math>H_0(S^2) \cong H_2(S^2) \cong \mathbb{Z}</math>, and all other homology groups are zero. The reduced homology groups with coefficients in <math>\mathbb{Z}</math> are as follows: <math>\tilde{H}_2(S^2) \cong \mathbb{Z}</math>, and all other reduced homology groups are zero.

Vipul: /* Topological space properties */

2011-07-21T23:35:34Z

Topological space properties

← Older revision		Revision as of 23:35, 21 July 2011
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	! Property !! Satisfied? !! Is the property a [[homotopy-invariant property of topological spaces]]? !! Explanation !! Corollary properties satisfied/dissatisfied		! Property !! Satisfied? !! Is the property a [[homotopy-invariant property of topological spaces]]? !! Explanation !! Corollary properties satisfied/dissatisfied
	\|-		\|-
	\| [[satisfies property::manifold]] \|\| Yes \|\| No \|\| Via stereographic projection, we see that the 2-sphere minus any point is homeomorphic to the [[Euclidean plane]]. Thus, we can give it an atlas with two charts, each chart obtained by removing a different point and mapping homeomorphically to the Euclidean plane. \|\| [[satisfies property::metrizable space]], [[satisfies property::second-countable space]], and all the separation axioms down from [[satisfies property::perfectly normal space]] and [[satisfies property::monotonically normal space]], including [[satisfies property::normal space\|normal]], [[satisfies property::completely regular space\|completely regular]], [[satisfies property::regular space\|regular]], [[satisfies property::Hausdorff space\|Hausdorff]], etc.		\| [[satisfies property::manifold]] \|\| Yes \|\| No \|\| Via stereographic projection, we see that the 2-sphere minus any point is homeomorphic to the [[Euclidean plane]]. Thus, we can give it an atlas with two charts, each chart obtained by removing a different point and mapping homeomorphically to the Euclidean plane. \|\| satisfies: [[satisfies property::metrizable space]], [[satisfies property::second-countable space]], and all the separation axioms down from [[satisfies property::perfectly normal space]] and [[satisfies property::monotonically normal space]], including [[satisfies property::normal space\|normal]], [[satisfies property::completely regular space\|completely regular]], [[satisfies property::regular space\|regular]], [[satisfies property::Hausdorff space\|Hausdorff]], etc.
	\|-		\|-
	\| [[satisfies property::path-connected space]] \|\| Yes \|\| Yes \|\| It is a union of two open subsets homeomorphic to the Euclidean plane (hence path-connected), and with non-empty intersection. Thus, it is path-connected. \|\| [[satisfies property::connected space]], [[satisfies property::connected manifold]], [[satisfies property::homogeneous space]] (via connected manifold, see [[connected manifold implies homogeneous]])		\| [[satisfies property::path-connected space]] \|\| Yes \|\| Yes \|\| It is a union of two open subsets homeomorphic to the Euclidean plane (hence path-connected), and with non-empty intersection. Thus, it is path-connected. \|\| satisfies: [[satisfies property::connected space]], [[satisfies property::connected manifold]], [[satisfies property::homogeneous space]] (via connected manifold, see [[connected manifold implies homogeneous]])
	\|-		\|-
	\| [[satisfies property::simply connected space]] \|\| Yes \|\| Yes \|\| Special case of [[n-sphere is simply connected for n greater than 1]]. Follows from [[union of two simply connected open subsets with path-connected intersection is simply connected]], which is a corollary of the [[Seifert-van Kampen theorem]] \|\| [[satisfies property::simply connected manifold]]		\| [[satisfies property::simply connected space]] \|\| Yes \|\| Yes \|\| Special case of [[n-sphere is simply connected for n greater than 1]]. Follows from [[union of two simply connected open subsets with path-connected intersection is simply connected]], which is a corollary of the [[Seifert-van Kampen theorem]] \|\| satisfies: [[satisfies property::simply connected manifold]]
	\|-		\|-
	\| [[dissatisfies property::rationally acyclic space]] \|\| No \|\| Yes \|\| The second homology group is isomorphic to the group of integers, hence it is nontrivial and has nontrivial torsion-free part. See [[homology of spheres]] \|\| [[dissatisfies property::acyclic space]], [[dissatisfies property::weakly contractible space]], [[dissatisfies property::contractible space]]		\| [[dissatisfies property::rationally acyclic space]] \|\| No \|\| Yes \|\| The second homology group is isomorphic to the group of integers, hence it is nontrivial and has nontrivial torsion-free part. See [[homology of spheres]] \|\| dissatisfies: [[dissatisfies property::acyclic space]], [[dissatisfies property::weakly contractible space]], [[dissatisfies property::contractible space]]
	\|-		\|-
	\| [[dissatisfies property::space with Euler characteristic zero]] \|\| No \|\| Yes \|\| The Euler characteristic is 2, see [[homology of spheres]] \|\|		\| [[dissatisfies property::space with Euler characteristic zero]] \|\| No \|\| Yes \|\| The Euler characteristic is 2, see [[homology of spheres]] \|\|
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	\| [[dissatisfies property::space with Euler characteristic one]] \|\| No \|\| Yes \|\| The Euler characteristic is 2, see [[homology of spheres]] \|\|		\| [[dissatisfies property::space with Euler characteristic one]] \|\| No \|\| Yes \|\| The Euler characteristic is 2, see [[homology of spheres]] \|\|
	\|-		\|-
	\| [[satisfies property::compact space]] \|\| Yes \|\| No \|\| Can be realized as a closed bounded subset of <math>\R^3</math> \|\| [[satisfies property::compact manifold]], [[satisfies property::compact polyhedron]], [[satisfies property::polyhedron]] (via compact manifold), [[satisfies property::compact Hausdorff space]], and all properties weaker than compactness		\| [[satisfies property::compact space]] \|\| Yes \|\| No \|\| Can be realized as a closed bounded subset of <math>\R^3</math> \|\| satisfies: [[satisfies property::compact manifold]], [[satisfies property::compact polyhedron]], [[satisfies property::polyhedron]] (via compact manifold), [[satisfies property::compact Hausdorff space]], and all properties weaker than compactness
	\|}		\|}

Vipul: /* Homotopy groups */

2011-07-21T23:26:35Z

Homotopy groups

← Older revision		Revision as of 23:26, 21 July 2011
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Value of <math>n</math> !! General name for <math>\~~pi_n~~</math> !! What is <math>\~~pi_n~~(S^2)</math>?		! Value of <math>k</math> !! General name for <math>\pi_k</math> !! What is <math>\pi_k(S^2)</math>?
	\|-		\|-
	\| 0 \|\| [[set of path components]] \|\| one-point set; so <math>S^2</math> is a [[satisfies property::path-connected space]]		\| 0 \|\| [[set of path components]] \|\| one-point set; so <math>S^2</math> is a [[satisfies property::path-connected space]]

Vipul: /* Homology-based invariants */

2011-06-18T16:59:52Z

Homology-based invariants

← Older revision		Revision as of 16:59, 18 June 2011
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Invariant !! General description !! Description of value for ~~spheres~~		! Invariant !! General description !! Description of value for sphere <math>S^n</math> !! Description of value for <math>S^2</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>.		\| [[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>. \|\| <math>b_0 = b_2 = 1</math>, all other <math>b_k = 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> \|\| <math>1 + x^2</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. \|\| 2
	\|}		\|}

Vipul: /* Topological space properties */

2011-04-28T00:03:59Z

Topological space properties

← Older revision		Revision as of 00:03, 28 April 2011
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	! Property !! Satisfied? !! Is the property a [[homotopy-invariant property of topological spaces]]? !! Explanation !! Corollary properties satisfied/dissatisfied		! Property !! Satisfied? !! Is the property a [[homotopy-invariant property of topological spaces]]? !! Explanation !! Corollary properties satisfied/dissatisfied
	\|-		\|-
	\| [[satisfies property::manifold]] \|\| Yes \|\| No \|\| Via stereographic projection, we see that the 2-sphere minus any point is homeomorphic to the [[Euclidean plane]]. Thus, we can give it an atlas with two charts, each chart obtained by removing a different point and mapping homeomorphically to the Euclidean plane. \|\| [[satisfies property::metrizable space]], [[satisfies property::second-countable space]], and all the separation axioms down from [[satisfies property::perfectly normal space]] and [[satisfies property::monotonically normal space]], including [[satisfies property::normal space\|normal]], [[satisfies property::completely regular\|completely regular]], [[satisfies property::regular space\|regular]], [[satisfies property::Hausdorff space\|Hausdorff]], etc.		\| [[satisfies property::manifold]] \|\| Yes \|\| No \|\| Via stereographic projection, we see that the 2-sphere minus any point is homeomorphic to the [[Euclidean plane]]. Thus, we can give it an atlas with two charts, each chart obtained by removing a different point and mapping homeomorphically to the Euclidean plane. \|\| [[satisfies property::metrizable space]], [[satisfies property::second-countable space]], and all the separation axioms down from [[satisfies property::perfectly normal space]] and [[satisfies property::monotonically normal space]], including [[satisfies property::normal space\|normal]], [[satisfies property::completely regular space\|completely regular]], [[satisfies property::regular space\|regular]], [[satisfies property::Hausdorff space\|Hausdorff]], etc.
	\|-		\|-
	\| [[satisfies property::path-connected space]] \|\| Yes \|\| Yes \|\| It is a union of two open subsets homeomorphic to the Euclidean plane (hence path-connected), and with non-empty intersection. Thus, it is path-connected. \|\| [[satisfies property::connected space]], [[satisfies property::connected manifold]], [[satisfies property::homogeneous space]] (via connected manifold, see [[connected manifold implies homogeneous]])		\| [[satisfies property::path-connected space]] \|\| Yes \|\| Yes \|\| It is a union of two open subsets homeomorphic to the Euclidean plane (hence path-connected), and with non-empty intersection. Thus, it is path-connected. \|\| [[satisfies property::connected space]], [[satisfies property::connected manifold]], [[satisfies property::homogeneous space]] (via connected manifold, see [[connected manifold implies homogeneous]])