# Difference between revisions of "Fundamental group"

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Perhaps the simplest object of study in algebraic topology is the '''fundamental group'''. | 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]], | + | Let <math>(X,x_0)</math> be a [[based topological space|based]], [[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>). Note that some authors will require <math>X</math> to be [[path-connected]]. 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>. |

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

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If <math>\circ,\cdot</math> share the same unit <math>e</math> (such that <math>a \cdot e = e \cdot a = a \circ e = e \circ a = a</math>) then <math>\cdot = \circ</math> and both are abelian. | If <math>\circ,\cdot</math> share the same unit <math>e</math> (such that <math>a \cdot e = e \cdot a = a \circ e = e \circ a = a</math>) then <math>\cdot = \circ</math> and both are abelian. | ||

+ | |||

+ | == Independence from base point == | ||

+ | |||

+ | At this point, one might wonder how significant the choice of base point, <math>x_0</math>, was. As it turns out, as long as <math>X</math> is path-connected, the choice of base point is irrelevant to the final group <math>\pi_1(X,x_0)</math>. | ||

+ | |||

+ | Indeed, pick consider any other base point <math>x_1</math>. As <math>X</math> is path connected, we can find a path <math>\alpha</math> from <math>x_0</math> to <math>x_1</math>. Let <math>\bar\alpha(t) = \alpha(1-t)</math> be the reverse path from <math>x_1</math> to <math>x_0</math>. For any <math>f\in\Omega(X,x_0)</math>, define <math>\varphi_\alpha = \bar\alpha\cdot f\cdot\alpha \in\Omega(X,x_1)</math> by | ||

+ | <cmath>\varphi_\alpha(f)(t) = (\bar\alpha\cdot f \cdot \alpha)(t) = \begin{cases} \bar\alpha(t) & 0\le t\le 1/3, \\ | ||

+ | f(3a-1) & 1/3\le t\le 2/3,\\ | ||

+ | \alpha(3t-2) & 2/3\le t\le 1. | ||

+ | \end{cases}</cmath> | ||

+ | One can now easily check that <math>\varphi_\alpha</math> is in fact a well-defined map <math>\pi_1(X,x_0)\to\pi_1(X,x_1)</math>, and furthermore, that it is a [[group homomorphism|homomorphism]]. Now we may similarly define the map <math>\varphi_{\bar\alpha}:\pi_1(X,x_1)\to\pi_1(X,x_0)</math> by <math>\varphi(g) = \alpha\cdot g\cdot\bar\alpha</math>. One can now easily verify that <math>\varphi_{\bar\alpha}</math> is the inverse of <math>\varphi_\alpha</math>. Thus <math>\varphi_\alpha</math> is an [[isomorphism]], so <math>\pi_1(X,x_0)\cong \pi_1(X,x_1)</math>. | ||

+ | |||

+ | Therefore (up to isomorphism), the group <math>\pi_1(X,x_0)</math> is independent of the choice of <math>x_0</math>. For this reason, we often just write <math>\pi_1(X)</math> for the fundamental of <math>X</math>. | ||

[[Category:Topology]] | [[Category:Topology]] | ||

[[Category:Algebraic Topology]] | [[Category:Algebraic Topology]] |

## Revision as of 01:17, 13 December 2009

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

Let be a based, topological space (that is, is a topological space, and is some point in ). Note that some authors will require to be path-connected. 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.

## Independence from base point

At this point, one might wonder how significant the choice of base point, , was. As it turns out, as long as is path-connected, the choice of base point is irrelevant to the final group .

Indeed, pick consider any other base point . As is path connected, we can find a path from to . Let be the reverse path from to . For any , define by One can now easily check that is in fact a well-defined map , and furthermore, that it is a homomorphism. Now we may similarly define the map by . One can now easily verify that is the inverse of . Thus is an isomorphism, so .

Therefore (up to isomorphism), the group is independent of the choice of . For this reason, we often just write for the fundamental of .