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]], [[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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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.
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== Independence from base point ==
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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>.
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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
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<cmath>\varphi_\alpha(f)(t) = (\bar\alpha\cdot f \cdot \alpha)(t) = \begin{cases} \bar\alpha(t) & 0\le t\le 1/3, \\
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f(3a-1) & 1/3\le t\le 2/3,\\
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\alpha(3t-2) & 2/3\le t\le 1.
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\end{cases}</cmath>
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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>.
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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 $(X,x_0)$ be a based, topological space (that is, $X$ is a topological space, and $x_0\in X$ is some point in $X$). Note that some authors will require $X$ to be path-connected. Now consider all possible "loops" on $X$ that start and end at $x_0$, i.e. all continuous functions $f:[0,1]\to X$ with $f(0)=f(1)=x_0$. Call this collection $\Omega(X,x_0)$ (the loop space of $X$). Now define an equivalence relation $\sim$ on $\Omega(X,x_0)$ by saying that $f\sim g$ if there is a (based) homotopy between $f$ and $g$ (that is, if there is a continuous function $F:[0,1]\times[0,1]\to X$ with $F(a,0)=f(a)$, $F(a,1)=g(a)$, and $F(0,b)=F(1,b)=x_0$). Now let $\pi_1(X,x_0)=\Omega(X,x_0)/\sim$ be the set of equivalence classes of $\Omega(X,x_0)$ under $\sim$.

Now define a binary operation $\cdot$ (called concatenation) on $\Omega(X,x_0)$ by $(g\cdot h)(a)=\begin{cases} g(2a) & 0\le a\le 1/2, \\ h(2a-1) & 1/2\le a\le 1.\end{cases}$ One can check that if $f\sim f'$ and $g\sim g'$ then $f\cdot g\sim f'\cdot g'$, and so $\cdot$ induces a well-defined binary operation on $\pi_1(X,x_0)$.

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

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 ${\mathbb{Z}}$, which is abelian.

More generally, if $X$ is an h-space, then $\pi_1(X)$ is abelian, for there is a second multiplication on $\pi_1(X)$ given by $(\alpha\beta)(t) = \alpha(t)\beta(t)$, which is "compatible" with the concatenation in the following respect:

We say that two binary operations $\circ, \cdot$ on a set $S$ are compatible if, for every $a,b,c,d \in S$, \[(a \circ b) \cdot (c \circ d) = (a \cdot c) \circ (b \cdot d).\]

If $\circ,\cdot$ share the same unit $e$ (such that $a \cdot e = e \cdot a = a \circ e = e \circ a = a$) then $\cdot = \circ$ and both are abelian.

Independence from base point

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

Indeed, pick consider any other base point $x_1$. As $X$ is path connected, we can find a path $\alpha$ from $x_0$ to $x_1$. Let $\bar\alpha(t) = \alpha(1-t)$ be the reverse path from $x_1$ to $x_0$. For any $f\in\Omega(X,x_0)$, define $\varphi_\alpha = \bar\alpha\cdot f\cdot\alpha \in\Omega(X,x_1)$ by \[\varphi_\alpha(f)(t) = (\bar\alpha\cdot f \cdot \alpha)(t) = \begin{cases} \bar\alpha(t) & 0\le t\le 1/3, \\  f(3a-1) & 1/3\le t\le 2/3,\\ \alpha(3t-2) & 2/3\le t\le 1. \end{cases}\] One can now easily check that $\varphi_\alpha$ is in fact a well-defined map $\pi_1(X,x_0)\to\pi_1(X,x_1)$, and furthermore, that it is a homomorphism. Now we may similarly define the map $\varphi_{\bar\alpha}:\pi_1(X,x_1)\to\pi_1(X,x_0)$ by $\varphi(g) = \alpha\cdot g\cdot\bar\alpha$. One can now easily verify that $\varphi_{\bar\alpha}$ is the inverse of $\varphi_\alpha$. Thus $\varphi_\alpha$ is an isomorphism, so $\pi_1(X,x_0)\cong \pi_1(X,x_1)$.

Therefore (up to isomorphism), the group $\pi_1(X,x_0)$ is independent of the choice of $x_0$. For this reason, we often just write $\pi_1(X)$ for the fundamental of $X$.

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