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One of the most elementary tools in algebraic topology is the [[fundamental group]], which is symbolized by <math>\pi_{1}(X) \ </math> for a given topological space <math>X \ </math>. Some details must be ironed out: for example, the investigation of loops which are very similar gives no insight into the structure of the space, so we work instead with sets of loops called homotopy classes.

One of the most elementary tools in algebraic topology is the [[fundamental group]], which is symbolized by <math>\pi_{1}(X) \ </math> for a given topological space <math>X \ </math>. Some details must be ironed out: for example, the investigation of loops which are very similar gives no insight into the structure of the space, so we work instead with sets of loops called homotopy classes.

 

 

 

The formal definition is as follows: Let <math>p \ </math> be a fixed point in <math>X \ </math>.  

The formal definition is as follows: Let <math>p \ </math> be a fixed point in <math>X \ </math>.  

Consider the space of all [[curve]]s <math>\gamma:[0,1]\rightarrow X \ </math> which begin and end at <math>p \ </math>.  

Consider the space of all [[curve]]s <math>\gamma:[0,1]\rightarrow X \ </math> which begin and end at <math>p \ </math>.  

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.

 

 

 

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  

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