Vipul: /* Definition */

2016-06-23T15:24:36Z

Definition

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	== Definition ==		== Definition ==

	A ''contractible manifold''' is a [[topological space]] satisfying the following equivalent conditions:		A '''contractible manifold''' is a [[topological space]] satisfying the following equivalent conditions:

	# It is both a [[contractible space]] and a [[manifold]].		# It is both a [[contractible space]] and a [[manifold]].

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2016-06-22T19:13:21Z

Created page with "{{topospace property conjunction|contractible space|manifold}} == Definition == A ''contractible manifold''' is a topological space satisfying the following equivalent c..."

New page

{{topospace property conjunction|contractible space|manifold}}

== Definition ==

A ''contractible manifold''' is a [[topological space]] satisfying the following equivalent conditions:

# It is both a [[contractible space]] and a [[manifold]].
# It is both a [[weakly contractible space]] and a [[manifold]].

== Examples ==

Any [[Euclidean space]] <math>\R^n</math> for <math>n</math> a nonnegative integer is a contractible manifold. For <math>n = 1</math> or <math>n = 2</math>, these are the only contractible manifolds up to homeomorphism. For <math>n = 3</math>, however, there exist contractible manifolds (in fact, contractible open subset of <math>\R^3</math>), such as the [[Whitehead manifold]], that are contractible but not homeomorphic to Euclidean space.

== 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::contractible space]] || || || || {{intermediate notions short|contractible space|contractible manifold}}
|-
| [[Stronger than::acyclic manifold]] || manifold that is an [[acyclic space]] || || || {{intermediate notions short|acyclic manifold|contractible manifold}}
|-
| [[Stronger than::simply connected manifold]] || manifold that is a [[simply connected space]] || || || {{intermediate notions short|simply connected manifold|contractible manifold}}
|-
| [[Stronger than::connected manifold]] || manifold that is a [[connected space]] || || || {{intermediate notions short|connected manifold|contractible manifold}}
|}
=== Opposite properties ===

[[Compact manifold]] is opposite in spirit: the only compact contractible manifold is the [[one-point space]]. To see this, note that:

* Any [[compact connected orientable manifold]] has top homology equal to <math>\mathbb{Z}</math>, making it non-contractible if its dimension is positive.
* If the manifold is not connected, it anyway cannot be contractible.
* If the manifold is not orientable, it has nontrivial fundamental group (in particular, the fundamental group admits <math>\mathbb{Z}/2\mathbb{Z}</math> as a quotient group) and is therefore not contractible.

Contractible manifold - Revision history

Vipul: /* Definition */

Vipul: Created page with "{{topospace property conjunction|contractible space|manifold}} == Definition == A ''contractible manifold''' is a topological space satisfying the following equivalent c..."