Editing Homotopy groups of spheres (section)

==={{math|π<sub>2</sub>(''S''<sup>1</sup>) {{=}} 0}}===
All the interesting cases of homotopy groups of spheres involve mappings from a higher-dimensional sphere onto one of lower dimension. Unfortunately, the only example which can easily be visualized is not interesting: there are no nontrivial mappings from the ordinary sphere to the circle. Hence, {{math|π<sub>2</sub>(''S''<sup>1</sup>) {{=}} 0}}. This is because {{math|''S''<sup>1</sup>}} has the real line as its [[universal cover]] which is contractible (it has the homotopy type of a point). In addition, because {{math|''S''<sup>2</sup>}} is simply connected, by the [[Homotopy lifting property|lifting criterion]],{{sfn|Hatcher|2002|p=61}} any map from {{math|''S''<sup>2</sup>}} to {{math|''S''<sup>1</sup>}} can be lifted to a map into the real line and the nullhomotopy descends to the downstairs space (via composition).