# Difference between revisions of "Fundamental group"

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− | Perhaps the simplest object of study in algebraic topology is the | + | Perhaps the simplest object of study in algebraic topology is the '''fundamental group'''. |

− | + | Let <math>(X,x_0)</math> be a [[based topological space|based]], [[path-connected]] [[topological space]] (that is, <math>X</math> is a topological space, and <math>x_0\in X</math> is some point in <math>X</math>). Now consider all possible "loops" on <math>X</math> that start and end at <math>x_0</math>, i.e. all [[continuous function]]s <math>f:[0,1]\to X</math> with <math>f(0)=f(1)=x_0</math>. Call this collection <math>\Omega(X,x_0)</math> (the '''loop space''' of <math>X</math>). Now define an [[equivalence relation]] <math>\sim</math> on <math>\Omega(X,x_0)</math> by saying that <math>f\sim g</math> if there is a (based) [[homotopy]] between <math>f</math> and <math>g</math> (that is, if there is a continuous function <math>F:[0,1]\times[0,1]\to X</math> with <math>F(a,0)=f(a)</math>, <math>F(a,1)=g(a)</math>, and <math>F(0,b)=F(1,b)=x_0</math>). Now let <math>\pi_1(X,x_0)=\Omega(X,x_0)/\sim</math> be the set of equivalence classes of <math>\Omega(X,x_0)</math> under <math>\sim</math>. | |

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+ | Now define a [[binary operation]] <math>\cdot</math> (called ''concatenation'') on <math>\Omega(X,x_0)</math> by | ||

+ | <math>(g\cdot h)(a)=\begin{cases} g(2a) & 0\le a\le 1/2, \\ h(2a-1) & 1/2\le a\le 1.\end{cases}</math> | ||

+ | One can check that if <math>f\sim f'</math> and <math>g\sim g'</math> then <math>f\cdot g\sim f'\cdot g'</math>, and so <math>\cdot</math> induces a well-defined binary operation on <math>\pi_1(X,x_0)</math>. | ||

+ | |||

+ | One can now check that the operation <math>\cdot</math> makes <math>\pi_1(X,x_0)</math> into a group. The identity element is just the constant loop <math>e(a) = x_0</math>, and the inverse of a loop <math>f</math> is just the loop <math>f</math> traversed in the opposite direction (i.e. the loop <math>\bar f(a) = f(1-a)</math>). We call <math>\pi_1(X,x_0)</math> the '''fundamental group''' of <math>X</math>. | ||

Note that the fundamental group is not in general [[abelian group|abelian]]. For example, the fundamental group of a figure eight is the [[free group]] on two [[generator]]s, which is not abelian. However, the fundamental group of a circle is <math>{\mathbb{Z}}</math>, which is abelian. | Note that the fundamental group is not in general [[abelian group|abelian]]. For example, the fundamental group of a figure eight is the [[free group]] on two [[generator]]s, which is not abelian. However, the fundamental group of a circle is <math>{\mathbb{Z}}</math>, which is abelian. |

## Revision as of 00:43, 13 December 2009

Perhaps the simplest object of study in algebraic topology is the **fundamental group**.

Let be a based, path-connected topological space (that is, is a topological space, and is some point in ). Now consider all possible "loops" on that start and end at , i.e. all continuous functions with . Call this collection (the **loop space** of ). Now define an equivalence relation on by saying that if there is a (based) homotopy between and (that is, if there is a continuous function with , , and ). Now let be the set of equivalence classes of under .

Now define a binary operation (called *concatenation*) on by
One can check that if and then , and so induces a well-defined binary operation on .

One can now check that the operation makes into a group. The identity element is just the constant loop , and the inverse of a loop is just the loop traversed in the opposite direction (i.e. the loop ). We call the **fundamental group** of .

Note that the fundamental group is not in general abelian. For example, the fundamental group of a figure eight is the free group on two generators, which is not abelian. However, the fundamental group of a circle is , which is abelian.

More generally, if is an h-space, then is abelian, for there is a second multiplication on given by , which is "compatible" with the concatenation in the following respect:

We say that two binary operations on a set are compatible if, for every ,

If share the same unit (such that ) then and both are abelian.