Editing Homotopy group (section)

== Homotopy groups and holes ==
A topological space has a [[Hole (topology)|''hole'']] with a ''d''-dimensional boundary if-and-only-if it contains a ''d''-dimensional sphere that cannot be shrunk continuously to a single point. This holds if-and-only-if there is a mapping  <math display="inline">S^d\to X</math> that is not homotopic to a [[constant function]]. This holds if-and-only-if the ''d''th homotopy group of ''X'' is not trivial. In short, ''X'' has a hole with a ''d''-dimensional boundary, if-and-only-if <math>\pi_d(X) \not \cong 0</math>.