Editing 3-sphere (section)

===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>}}
|}