Editing Homotopy groups of spheres (section)

=== {{mvar|n}}-sphere ===
An ordinary [[sphere]] in three-dimensional space—the surface, not the solid ball—is just one example of what a sphere means in topology. [[Geometry]] defines a sphere rigidly, as a shape. Here are some alternatives.
* '''Implicit surface''': {{math|''x''{{su|lh=1|b=0|p=2}} + ''x''{{su|lh=1|b=1|p=2}} + ''x''{{su|lh=1|b=2|p=2}} {{=}} 1}}
: This is the set of points in 3-dimensional [[Euclidean space]] found exactly one unit away from the origin. It is called the 2-sphere, {{math|''S''<sup>2</sup>}}, for reasons given below. The same idea applies for any [[dimension]] {{mvar|n}}; the equation {{math|''x''{{su|lh=1|b=0|p=2}} + ''x''{{su|lh=1|b=1|p=2}} + ⋯ + ''x''{{su|lh=1|b=''n''|p=2}} {{=}} 1}} produces the [[n-sphere|{{mvar|n}}-sphere]] as a geometric object in ({{math|''n'' + 1}})-dimensional space. For example, the 1-sphere {{math|''S''<sup>1</sup>}} is a [[circle]].{{sfn|Hatcher|2002|p=xii}}
* '''Disk with collapsed rim''': written in topology as {{math|''D''<sup>2</sup>/''S''<sup>1</sup>}}
: This construction moves from geometry to pure topology. The [[unit disk|disk]] {{math|''D''<sup>2</sup>}} is the region contained by a circle, described by the inequality {{math|''x''{{su|lh=1|b=0|p=2}} + ''x''{{su|lh=1|b=1|p=2}} ≤ 1}}, and its rim (or "[[boundary (topology)|boundary]]") is the circle {{math|''S''<sup>1</sup>}}, described by the equality {{math|''x''{{su|lh=1|b=0|p=2}} + ''x''{{su|lh=1|b=1|p=2}} {{=}} 1}}. If a [[balloon]] is punctured and spread flat it produces a disk; this construction repairs the puncture, like pulling a drawstring. The [[Slash (punctuation)|slash]], pronounced "modulo", means to take the topological space on the left (the disk) and in it join together as one all the points on the right (the circle). The region is 2-dimensional, which is why topology calls the resulting topological space a 2-sphere. Generalized, {{math|''D''<sup>''n''</sup>/''S''<sup>''n''−1</sup>}} produces {{math|''S''<sup>''n''</sup>}}. For example, {{math|''D''<sup>1</sup>}} is a [[line segment]], and the construction joins its ends to make a circle. An equivalent description is that the boundary of an {{mvar|n}}-dimensional disk is glued to a point, producing a [[CW complex]].{{sfn|Hatcher|2002|loc=Example 0.3, p. 6}}
* '''Suspension of equator''': written in topology as {{math|Σ''S''<sup>1</sup>}}
: This construction, though simple, is of great theoretical importance. Take the circle {{math|''S''<sup>1</sup>}} to be the [[equator]], and sweep each point on it to one point above (the North Pole), producing the northern hemisphere, and to one point below (the South Pole), producing the southern hemisphere. For each positive integer {{mvar|n}}, the {{mvar|n}}-sphere {{math|''x''{{su|lh=1|b=0|p=2}} + ''x''{{su|lh=1|b=1|p=2}} + ⋯ + ''x''{{su|lh=1|b=''n''|p=2}} {{=}} 1}} has as equator the ({{math|''n'' &minus; 1}})-sphere {{math|''x''{{su|lh=1|b=0|p=2}} + ''x''{{su|lh=1|b=1|p=2}} + ⋯ + ''x''{{su|lh=1|b=''n''−1|p=2}} {{=}} 1}}, and the suspension {{math|Σ''S''<sup>''n''−1</sup>}} produces {{math|''S''<sup>''n''</sup>}}.{{sfn|Hatcher|2002|p=129}}

Some theory requires selecting a fixed point on the sphere, calling the pair {{math|(sphere, point)}} a ''[[pointed space|pointed sphere]]''. For some spaces the choice matters, but for a sphere all points are equivalent so the choice is a matter of convenience.{{sfn|Hatcher|2002|p=28}} For spheres constructed as a repeated suspension, the point {{math|(1, 0, 0, ..., 0)}}, which is on the equator of all the levels of suspension, works well; for the disk with collapsed rim, the point resulting from the collapse of the rim is another obvious choice.