Editing Homotopy groups of spheres (section)

==={{math|π<sub>1</sub>(''S''<sup>1</sup>) {{=}} Z}}===
[[Image:Fundamental group of the circle.svg|300px|thumb|Elements of {{math|π<sub>1</sub>(''S''<sup>1</sup>)}}]]

The simplest case concerns the ways that a circle (1-sphere) can be wrapped around another circle. This can be visualized by wrapping a [[rubber band]] around one's finger: it can be wrapped once, twice, three times and so on.  The wrapping can be in either of two directions, and wrappings in opposite directions will cancel out after a deformation. The homotopy group {{math|π<sub>1</sub>(''S''<sup>1</sup>)}} is therefore an [[infinite cyclic group]], and is [[isomorphic]] to the group {{math|Z}} of [[integer]]s under addition: a homotopy class is identified with an integer by counting the number of times a mapping in the homotopy class wraps around the circle. This integer can also be thought of as the [[winding number]] of a loop around the [[origin (mathematics)|origin]] in the [[plane (mathematics)|plane]].{{sfn|Hatcher|2002|p=29}}

The identification (a [[group isomorphism]]) of the homotopy group with the integers is [[abuse of notation|often written]] as an equality: thus {{math|π<sub>1</sub>(''S''<sup>1</sup>) {{=}} Z}}.<ref>See, e.g., {{harvnb|Homotopy type theory|2013}}, Section 8.1, "<math display=inline>\pi_1(S^1)</math>".</ref>