Homology sphere - Revision history

Vipul: /* Definition */

2016-06-22T06:47:13Z

Definition

← Older revision		Revision as of 06:47, 22 June 2016
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	== Definition ==		== Definition ==

	Suppose <math>n</math> is a natural number. A '''homology <math>n</math>-sphere'''' is a <math>n</math>-dimensional [[manifold]] whose [[homology group]]s (over the ring of integers <math>\mathbb{Z}</math>) match those of the [[sphere]]. Specifically, a [[manifold]] <math>M</math> is a homology sphere if its homology groups are as follows:		Suppose <math>n</math> is a natural number. A '''homology <math>n</math>-sphere''' is a <math>n</math>-dimensional [[manifold]] whose [[homology group]]s (over the ring of integers <math>\mathbb{Z}</math>) match those of the [[sphere]]. Specifically, a [[manifold]] <math>M</math> is a homology sphere if its homology groups are as follows:

	<math>H_i(M; \mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z} & i = 0,n \\ 0 & i \ne 0, n \end{array}\right.</math>		<math>H_i(M; \mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z} & i = 0,n \\ 0 & i \ne 0, n \end{array}\right.</math>

Vipul: /* Relation with other properties */

2016-06-22T06:44:18Z

Relation with other properties

← Older revision		Revision as of 06:44, 22 June 2016
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Stronger than::rational homology sphere]] \|\| manifold whose homology groups over rationals match those of a sphere of the same dimension \|\| \|\| \|\| {{intermediate notions short\|rational homology sphere\|homology sphere}}
	\|-		\|-
	\| [[Stronger than::compact connected orientable manifold]] \|\| \|\| \|\| \|\| {{intermediate notions short\|compact connected orientable manifold\|homology sphere}}		\| [[Stronger than::compact connected orientable manifold]] \|\| \|\| \|\| \|\| {{intermediate notions short\|compact connected orientable manifold\|homology sphere}}

Vipul at 06:43, 22 June 2016

2016-06-22T06:43:31Z

← Older revision		Revision as of 06:43, 22 June 2016
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	Suppose <math>n</math> is a natural number. A '''homology <math>n</math>-sphere'''' is a <math>n</math>-dimensional [[manifold]] whose [[homology group]]s (over the ring of integers <math>\mathbb{Z}</math>) match those of the [[sphere]]. Specifically, a [[manifold]] <math>M</math> is a homology sphere if its homology groups are as follows:		Suppose <math>n</math> is a natural number. A '''homology <math>n</math>-sphere'''' is a <math>n</math>-dimensional [[manifold]] whose [[homology group]]s (over the ring of integers <math>\mathbb{Z}</math>) match those of the [[sphere]]. Specifically, a [[manifold]] <math>M</math> is a homology sphere if its homology groups are as follows:

	<math>H_i(M; \mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z} & i = 0,n \\ 0 & i \ne 0, n\right.</math>		<math>H_i(M; \mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z} & i = 0,n \\ 0 & i \ne 0, n \end{array}\right.</math>

	Note that it is important to explicitly specify that the manifold is <math>n</math>-dimensional, otherwise a [[cylinder]] over a sphere would satisfy the definition.		Note that it is important to explicitly specify that the manifold is <math>n</math>-dimensional, otherwise a [[cylinder]] over a sphere would satisfy the definition.

Vipul: Created page with "== Definition == Suppose $n$ is a natural number. A '''homology $n$ -sphere'''' is a $n$ -dimensional manifold whose homology groups (o..."

2016-06-22T06:43:16Z

Created page with "== Definition == Suppose <math>n</math> is a natural number. A '''homology <math>n</math>-sphere'''' is a <math>n</math>-dimensional manifold whose homology groups (o..."

New page

== Definition ==

Suppose <math>n</math> is a natural number. A '''homology <math>n</math>-sphere'''' is a <math>n</math>-dimensional [[manifold]] whose [[homology group]]s (over the ring of integers <math>\mathbb{Z}</math>) match those of the [[sphere]]. Specifically, a [[manifold]] <math>M</math> is a homology sphere if its homology groups are as follows:

<math>H_i(M; \mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z} & i = 0,n \\ 0 & i \ne 0, n\right.</math>

Note that it is important to explicitly specify that the manifold is <math>n</math>-dimensional, otherwise a [[cylinder]] over a sphere would satisfy the definition.

Note that we exclude the case <math>n = 0</math> from consideration.

== Examples ==

For any positive integer <math>n</math>, the <math>n</math>-sphere <math>S^n</math> is a homology sphere.

For <math>n = 3</math>, the [[Poincare homology sphere]], obtained as the quotient of the [[3-sphere]] by the binary icosahedral group (that is isomorphic to <math>SL(2,5)</math>) is a homology sphere.

== Facts ==

=== Complement of a point ===

The complement of any point in a homology sphere is an [[acyclic space]]. In particular, when the homology sphere is ''not'' a sphere (and specifically, its [[fundamental group]] is a nontrivial [[perfect group]]) then the complement of a point in it is acyclic but ''not'' [[weakly contractible space|weakly contractible]].

=== Suspension ===

* The suspension of any homology 3-sphere that is not homeomorphic to the 3-sphere is a [[homology manifold]] that is not a manifold.

== Relation with other properties ==

=== Weaker properties ===

{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::compact connected orientable manifold]] || || || || {{intermediate notions short|compact connected orientable manifold|homology sphere}}
|-
| For <math>n > 1</math>, [[space with perfect fundamental group]] || || || ||
|-
| For odd <math>n</math>, [[space with Euler characteristic zero]] || || || ||
|}

Homology sphere - Revision history

Vipul: /* Definition */

Vipul: /* Relation with other properties */

Vipul at 06:43, 22 June 2016

Vipul: Created page with "== Definition == Suppose n is a natural number. A '''homology n-sphere'''' is a n-dimensional manifold whose homology groups (o..."

Vipul: Created page with "== Definition == Suppose $n$ is a natural number. A '''homology $n$ -sphere'''' is a $n$ -dimensional manifold whose homology groups (o..."