Editing 3-sphere (section)

==Properties==

===Elementary properties===
The 3-dimensional surface volume of a 3-sphere of radius {{mvar|r}} is
:<math>SV=2\pi^2 r^3 \,</math>
while the 4-dimensional hypervolume (the content of the 4-dimensional region, or ball, bounded by the 3-sphere) is
:<math>H=\frac{1}{2} \pi^2 r^4.</math>

Every non-empty intersection of a 3-sphere with a three-dimensional [[hyperplane]] is a 2-sphere (unless the hyperplane is tangent to the 3-sphere, in which case the intersection is a single point). As a 3-sphere moves through a given three-dimensional hyperplane, the intersection starts out as a point, then becomes a growing 2-sphere that reaches its maximal size when the hyperplane cuts right through the "equator" of the 3-sphere. Then the 2-sphere shrinks again down to a single point as the 3-sphere leaves the hyperplane.

In a given three-dimensional hyperplane, a 3-sphere can rotate about an "equatorial plane" (analogous to a 2-sphere rotating about a central axis), in which case it appears to be a 2-sphere whose size is constant.

===Topological properties===
A 3-sphere is a [[Compact space|compact]], [[connected space|connected]], 3-dimensional [[manifold]] without boundary. It is also [[simply connected]]. What this means, in the broad sense, is that any loop, or circular path, on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The [[Poincaré conjecture]], proved in 2003 by [[Grigori Perelman]], provides that the 3-sphere is the only three-dimensional manifold (up to [[homeomorphism]]) with these properties.

The 3-sphere is homeomorphic to the [[one-point compactification]] of {{math|'''R'''<sup>3</sup>}}. In general, any [[topological space]] that is homeomorphic to the 3-sphere is called a '''topological 3-sphere'''.

The [[homology group]]s of the 3-sphere are as follows: {{math|H<sub>0</sub>(''S''<sup>3</sup>, '''Z''')}} and {{math|H<sub>3</sub>(''S''<sup>3</sup>, '''Z''')}} are both [[infinite cyclic]], while {{math|1=H<sub>''i''</sub>(''S''<sup>3</sup>, '''Z''') = {}{{null}}}} for all other indices {{mvar|i}}. Any topological space with these homology groups is known as a [[homology sphere|homology 3-sphere]]. Initially [[Henri Poincaré|Poincaré]] conjectured that all homology 3-spheres are homeomorphic to {{math|''S''<sup>3</sup>}}, but then he himself constructed a non-homeomorphic one, now known as the [[Poincaré homology sphere]]. Infinitely many homology spheres are now known to exist. For example, a [[Dehn filling]] with slope {{math|{{sfrac|1|''n''}}}} on any [[knot theory|knot]] in the 3-sphere gives a homology sphere; typically these are not homeomorphic to the 3-sphere.

As to the [[homotopy groups]], we have {{math|1=π<sub>1</sub>(''S''<sup>3</sup>) = π<sub>2</sub>(''S''<sup>3</sup>) = {}{{null}}}} and {{math|π<sub>3</sub>(''S''<sup>3</sup>)}} is infinite cyclic. The higher-homotopy groups ({{math|''k'' ≥ 4}}) are all [[finite abelian group|finite abelian]] but otherwise follow no discernible pattern. For more discussion see [[homotopy groups of spheres]].
{| class="wikitable" style="text-align: center; margin: auto;"
|+Homotopy groups of {{math|''S''<sup>3</sup>}}
|-
| {{mvar|k}}
| 0 || 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16
|-
| {{math|π<sub>''k''</sub>(''S''<sup>3</sup>)}}
| 0 || 0 || 0 || {{math|'''Z'''}} || {{math|'''Z'''<sub>2</sub>}} || {{math|'''Z'''<sub>2</sub>}} || {{math|'''Z'''<sub>12</sub>}} || {{math|'''Z'''<sub>2</sub>}} || {{math|'''Z'''<sub>2</sub>}} || {{math|'''Z'''<sub>3</sub>}} || {{math|'''Z'''<sub>15</sub>}} || {{math|'''Z'''<sub>2</sub>}} || {{math|'''Z'''<sub>2</sub>⊕'''Z'''<sub>2</sub>}} || {{math|'''Z'''<sub>12</sub>⊕'''Z'''<sub>2</sub>}} || {{math|'''Z'''<sub>84</sub>⊕'''Z'''<sub>2</sub>⊕'''Z'''<sub>2</sub>}} || {{math|'''Z'''<sub>2</sub>⊕'''Z'''<sub>2</sub>}} || {{math|'''Z'''<sub>6</sub>}}
|}

===Geometric properties===
The 3-sphere is naturally a [[smooth manifold]], in fact, a closed [[embedded submanifold]] of {{math|'''R'''<sup>4</sup>}}. The [[Euclidean metric]] on {{math|'''R'''<sup>4</sup>}} induces a [[metric tensor|metric]] on the 3-sphere giving it the structure of a [[Riemannian manifold]]. As with all spheres, the 3-sphere has constant positive [[sectional curvature]] equal to {{math|{{sfrac|1|''r''<sup>2</sup>}}}} where {{mvar|r}} is the radius.

Much of the interesting geometry of the 3-sphere stems from the fact that the 3-sphere has a natural [[Lie group]] structure given by quaternion multiplication (see the section below on [[#Group structure|group structure]]). The only other spheres with such a structure are the 0-sphere and the 1-sphere (see [[circle group]]).

Unlike the 2-sphere, the 3-sphere admits nonvanishing [[vector field]]s ([[section (fiber bundle)|sections]] of its [[tangent bundle]]). One can even find three linearly independent and nonvanishing vector fields. These may be taken to be any left-invariant vector fields forming a basis for the [[Lie algebra]] of the 3-sphere. This implies that the 3-sphere is [[Parallelizable manifold|parallelizable]]. It follows that the tangent bundle of the 3-sphere is [[trivial bundle|trivial]]. For a general discussion of the number of linear independent vector fields on a {{mvar|n}}-sphere, see the article [[vector fields on spheres]].

There is an interesting [[Group action (mathematics)|action]] of the [[circle group]] {{math|'''T'''}} on {{math|''S''<sup>3</sup>}} giving the 3-sphere the structure of a [[principal circle bundle]] known as the [[Hopf bundle]]. If one thinks of {{math|''S''<sup>3</sup>}} as a subset of {{math|'''C'''<sup>2</sup>}}, the action is given by
:<math>(z_1,z_2)\cdot\lambda = (z_1\lambda,z_2\lambda)\quad \forall\lambda\in\mathbb T</math>.
The [[orbit space]] of this action is homeomorphic to the two-sphere {{math|''S''<sup>2</sup>}}. Since {{math|''S''<sup>3</sup>}} is not homeomorphic to {{math|''S''<sup>2</sup> × ''S''<sup>1</sup>}}, the Hopf bundle is nontrivial.