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Higher homotopy groups can be defined, and are written as <math>\pi_n(X) \ </math>.  These are defined in just the same way, except we consider maps <math>\gamma:[0,1]^n \rightarrow X \ </math> instead.

Higher homotopy groups can be defined, and are written as <math>\pi_n(X) \ </math>.  These are defined in just the same way, except we consider maps <math>\gamma:[0,1]^n \rightarrow X \ </math> instead.

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A very important example is the topological space <math>\mathbb{S}^1 \ </math>, the circle. It can be shown that the fundamental group of <math>\mathbb{S}^1 \ </math> is isomorphic to the additive group of [[integer]]s <math>\mathbb{Z}</math>. Other examples include the torus, <math>\pi_1(\mathbb{T}^n) \ </math>, which is the free group on <math>n \ </math> generators, the sphere  

A very important example is the topological space <math>\mathbb{S}^1 \ </math>, the circle. It can be shown that the fundamental group of <math>\mathbb{S}^1 \ </math> is isomorphic to the additive group of [[integer]]s <math>\mathbb{Z}</math>. Other examples include the torus, <math>\pi_1(\mathbb{T}^n) \ </math>, which is the free abelian group on <math>n \ </math> generators, the sphere  

<math>\pi_m(\mathbb{S}^n) = \begin{cases} \left\{{e}\right\}, & \mbox{if }  m< n  \\ \mathbb{Z},  & \mbox{if } m=n \end{cases} \ </math>

<math>\pi_m(\mathbb{S}^n) = \begin{cases} \left\{{e}\right\}, & \mbox{if }  m< n  \\ \mathbb{Z},  & \mbox{if } m=n \end{cases} \ </math>