Difference between revisions of "The fundamental group"

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'''Requires: ''' [[Paths and loops in a topological space]] and [[Homotopic paths]]<ref>Introduction to topology - lecture notes nov 2013 - David Mond</ref>
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{{Refactor notice|grade=A|msg=I cannot believe it's been 15 months and this still isn't complete!
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* Started refactoring [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 19:55, 1 November 2016 (UTC)}}
 
==Definition==
 
==Definition==
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Given a [[Topological space|topological space]] {{M|X}} and a point {{M|x_0\in X}}
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Let {{Top.|X|J}} be a [[topological space]] {{M|\text{Loop}(X,b)\subseteq C(I,X)}} and consider the [[relation]] of [[path homotopic maps|path homotopic maps, {{M|\big((\cdot)\simeq(\cdot)\ (\text{rel }\{0,1\})\big)}}]] on {{M|C(I,X)}} and restricted to {{M|\text{Loop}(X,b)}}, then:
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{{Todo|Fundamental group}}
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* {{M|1=\pi_1(X,b):=\frac{\text{Loop}(X,b)}{\big((\cdot)\simeq(\cdot)\ (\text{rel }\{0,1\})\big)} }} has a [[group]] structure, with the [[group operation]] being:
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** {{M|:[\ell_1]\cdot[\ell_2]\mapsto[\ell_1*\ell_2]}} where {{M|\ell_1*\ell_2}} denotes the [[loop concatenation]] of {{M|\ell_1,\ell_2\in\text{Loop}(X,b)}}.
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==Proof of claims==
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{{Begin Inline Theorem}}
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[[Proof that the fundamental group is actually a group|Outline of proof that {{M|\pi_1(X,b)}} admits a group structure with {{M|\big(:([\ell_1],[\ell_2])\mapsto[\ell_1*\ell_2]\big)}} as the operation]]
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{{Begin Inline Proof}}{{:Proof that the fundamental group is actually a group/Outline}}
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{{End Proof}}{{End Theorem}}
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{{Begin Inline Theorem}}
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[[Proof that the fundamental group is actually a group|Proof that {{M|\pi_1(X,b)}} admits a group structure with {{M|\big(:([\ell_1],[\ell_2])\mapsto[\ell_1*\ell_2]\big)}} as the operation]]
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{{Begin Inline Proof}}{{:Proof that the fundamental group is actually a group/Proof}}
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{{End Proof}}{{End Theorem}}
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==References==
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<references/>
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{{Definition|Topology|Homotopy Theory}}
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=OLD PAGE=
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'''Requires: ''' [[Paths and loops in a topological space]] and [[Homotopic paths]]
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==Definition==
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Given a [[Topological space|topological space]] {{M|X}} and a point {{M|x_0\in X}} the fundamental group is<ref>Introduction to Topology - Second Edition - Theodore W. Gamelin and Rober Everist Greene</ref>
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* <math>\pi_1(X,x_0)</math> denotes the set of [[Homotopy class|homotopy classes]] of [[Paths and loops in a topological space|loops]] based at {{M|x_0}}
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: forms a [[Group|group]] under the operation of multiplication of the homotopy classes.
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{{Begin Theorem}}
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Theorem: {{M|\pi_1(X,x_0)}} with the binary operation {{M|*}} forms a [[Group|group]]<ref>Introduction to topology - lecture notes nov 2013 - David Mond</ref>
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{{Begin Proof}}
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* Identity element
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* Inverses
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* Association
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See [[Homotopy class]] for these properties
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{{Todo|Mond p30}}
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{{End Proof}}
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{{End Theorem}}
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==See also==
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* [[Homotopy class]]
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* [[Homotopic paths]]
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* [[Paths and loops in a topological space]]
  
 
==References==
 
==References==

Latest revision as of 16:10, 4 November 2016

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  • Started refactoring Alec (talk) 19:55, 1 November 2016 (UTC)

Definition

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space [ilmath]\text{Loop}(X,b)\subseteq C(I,X)[/ilmath] and consider the relation of path homotopic maps, [ilmath]\big((\cdot)\simeq(\cdot)\ (\text{rel }\{0,1\})\big)[/ilmath] on [ilmath]C(I,X)[/ilmath] and restricted to [ilmath]\text{Loop}(X,b)[/ilmath], then:

  • [ilmath]\pi_1(X,b):=\frac{\text{Loop}(X,b)}{\big((\cdot)\simeq(\cdot)\ (\text{rel }\{0,1\})\big)}[/ilmath] has a group structure, with the group operation being:
    • [ilmath]:[\ell_1]\cdot[\ell_2]\mapsto[\ell_1*\ell_2][/ilmath] where [ilmath]\ell_1*\ell_2[/ilmath] denotes the loop concatenation of [ilmath]\ell_1,\ell_2\in\text{Loop}(X,b)[/ilmath].

Proof of claims

Outline of proof that [ilmath]\pi_1(X,b)[/ilmath] admits a group structure with [ilmath]\big(:([\ell_1],[\ell_2])\mapsto[\ell_1*\ell_2]\big)[/ilmath] as the operation

[ilmath]\xymatrix{ \Omega(X,b)\times\Omega(X,b) \ar@2{->}[d]_{(\pi,\pi)} \ar[rr]^-{*} \ar@/^3.5ex/[drr]^(.75){\pi\circ *} & & \Omega(X,b) \ar[d]^{\pi} \\ \pi_1(X,b)\times\pi_1(X,b) \ar@{.>}[rr]_-{\overline{*} } & & \pi_1(X,b) }[/ilmath]
Factoring [ilmath]*[/ilmath] (loop concatenation) setup
Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]b\in X[/ilmath] be given. Then [ilmath]\Omega(X,b)[/ilmath] is the set of all loops based at [ilmath]b[/ilmath]. Let [ilmath]{\small(\cdot)}\simeq{\small(\cdot)}\ (\text{rel }\{0,1\})[/ilmath] denote the relation of end-point-preserving homotopy on [ilmath]C([0,1],X)[/ilmath] - the set of all paths in [ilmath]X[/ilmath] - but considered only on the subset of [ilmath]C([0,1],X)[/ilmath], [ilmath]\Omega(X,b)[/ilmath].

Then we define: [math]\pi_1(X,b):=\frac{\Omega(X,b)}{\big({\small(\cdot)}\simeq{\small(\cdot)}\ (\text{rel }\{0,1\})\big)}[/math], a standard quotient by an equivalence relation.

Consider the binary function: [ilmath]*:\Omega(X,b)\times\Omega(X,b)\rightarrow\Omega(X,b)[/ilmath] defined by loop concatenation, or explicitly:

  • [ilmath]*:(\ell_1,\ell_2)\mapsto\left(\ell_1*\ell_2:[0,1]\rightarrow X\text{ given by }\ell_1*\ell_2:t\mapsto\left\{\begin{array}{lr}\ell_1(2t) & \text{for }t\in[0,\frac{1}{2}]\\ \ell_2(2t-1) & \text{for }t\in[\frac{1}{2},1]\end{array}\right.\right)[/ilmath]
    • notice that [ilmath]t=\frac{1}{2}[/ilmath] is in both parts, this is a nod to the pasting lemma

We now have the situation on the right. Note that:

  • [ilmath](\pi,\pi):\Omega(X,b)\times\Omega(X,b)\rightarrow\pi_1(X,b)\times\pi_1(X,b)[/ilmath] is just [ilmath]\pi[/ilmath] applied to an ordered pair, [ilmath](\pi,\pi):(\ell_1,\ell_2)\mapsto([\ell_1],[\ell_2])[/ilmath]


In order to factor [ilmath](\pi\circ *)[/ilmath] through [ilmath](\pi,\pi)[/ilmath] we require (as per the factor (function) page) that:

  • [ilmath]\forall(\ell_1,\ell_2),(\ell_1',\ell_2')\in\Omega(X,b)\times\Omega(X,b)\big[\big((\pi,\pi)(\ell_1,\ell_2)=(\pi,\pi)(\ell_1',\ell_2')\big)\implies\big(\pi(\ell_1*\ell_2)=\pi(\ell_1'*\ell_2')\big)\big][/ilmath], this can be written better using our standard notation:
    • [ilmath]\forall\ell_1,\ell_2,\ell_1',\ell_2'\in\Omega(X,b)\big[\big(([\ell_1],[\ell_2])=([\ell_1'],[\ell_2'])\big)\implies\big([\ell_1*\ell_2]=[\ell_1'*\ell_2']\big)\big][/ilmath]


Then we get (just by applying the function factorisation theorem):

  • [ilmath]\overline{*}:\pi_1(X,b)\times\pi_1(X,b)\rightarrow\pi_1(X,b)[/ilmath] given (unambiguously) by [ilmath]\overline{*}:([\ell_1],[\ell_2])\mapsto[\ell_1*\ell_2][/ilmath] or written more nicely as:
    • [ilmath][\ell_1]\overline{*}[\ell_2]:=[\ell_1*\ell_2][/ilmath]


Lastly we show [ilmath](\pi_1(X,b),\overline{*})[/ilmath] forms a group

Proof that [ilmath]\pi_1(X,b)[/ilmath] admits a group structure with [ilmath]\big(:([\ell_1],[\ell_2])\mapsto[\ell_1*\ell_2]\big)[/ilmath] as the operation

We wish to show that the set [ilmath]\pi_1(X,b):=\frac{\Omega(X,b)}{\big({\small(\cdot)}\simeq{\small (\cdot)}\ (\text{rel }\{0,1\})\big)}[/ilmath] is actually a group with the operation [ilmath]\overline{*} [/ilmath] as described in the outline.

  1. Factoring:
    • Setup:
      • [ilmath]*:\Omega(X,b)\times\Omega(X,b)\rightarrow\Omega(X,b)[/ilmath] - the operation of loop concatenation - [ilmath]*:(\ell_1,\ell_2)\mapsto\left(\ell_1*\ell_2:I\rightarrow X\text{ by }\ell_1*\ell_2:t\mapsto\left\{\begin{array}{lr}\ell_1(2t) & \text{for }t\in[0,\frac{1}{2}]\\ \ell(2t-1) & \text{for }t\in[\frac{1}{2},1]\end{array}\right.\right)[/ilmath]
      through
      • [ilmath](p,p):\Omega(X,b)\times\Omega(X,b)\rightarrow\pi_1(X,b)\times\pi_1(X,b)[/ilmath] by [ilmath](p,p):(\ell_1,\ell_2)\mapsto(p(\ell_1),p(\ell_2))[/ilmath]
        • where [ilmath]p:\Omega(X,b)\rightarrow\pi_1(X,b)[/ilmath] is the canonical projection of the equivalence relation. As such we may say:
        • [ilmath](p,p)[/ilmath] is given by by [ilmath](p,p):(\ell_1,\ell_2)\mapsto([\ell_1],[\ell_2])[/ilmath] instead
      • We must show:
        • [ilmath]\forall\ell_1,\ell_2,\ell_1',\ell_2'\in\Omega(X,b)\left[\big([\ell_1]=[\ell_1']\wedge[\ell_2]=[\ell_2']\big)\implies\big([\ell_1*\ell_2]=[\ell_1'*\ell_2']\big)\right][/ilmath][Note 1]
    • Proof:
      • Let [ilmath]\ell_1,\ell_2,\ell_1',\ell_2'\in\Omega(X,b)[/ilmath] be given
        • Suppose that [ilmath]\neg([\ell_1]=[\ell_1']\wedge[\ell_2]=[\ell_2'])[/ilmath] holds, then by the nature of logical implication we're done, as we do not care about the RHS's truth or falsity in this case
        • Suppose that [ilmath][\ell_1]=[\ell_1']\wedge[\ell_2]=[\ell_2'][/ilmath] holds. We must show that in this case we have [ilmath][\ell_1*\ell_2]=[\ell_1'*\ell_2'][/ilmath]
      • Since [ilmath]\ell_1,\ell_2,\ell_1'[/ilmath] and [ilmath]\ell_2'[/ilmath] were arbitrary this holds for all.
    • Conclusion
      • We obtain [ilmath]\overline{*}:\pi_1(X,b)\times\pi_1(X,b)\rightarrow\pi_1(X,b)[/ilmath] given unambiguously by:
        • [ilmath]\overline{*}:([\ell_1],[\ell_2])\mapsto[\ell_1*\ell_2][/ilmath]
    • Thus the group operation is:
      • [ilmath][\ell_1]\overline{*}[\ell_2]:=[\ell_1*\ell_2][/ilmath]
  2. Associativity of the operation [ilmath]\overline{*} [/ilmath]
  3. Existence of an identity element in [ilmath](\pi_1(X,b),\overline{*})[/ilmath]
  4. For each element of [ilmath]\pi_1(X,b)[/ilmath] the existence of an inverse element in [ilmath](\pi_1(X,b),\overline{*})[/ilmath]
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References


OLD PAGE

Requires: Paths and loops in a topological space and Homotopic paths

Definition

Given a topological space [ilmath]X[/ilmath] and a point [ilmath]x_0\in X[/ilmath] the fundamental group is[1]

forms a group under the operation of multiplication of the homotopy classes.

Theorem: [ilmath]\pi_1(X,x_0)[/ilmath] with the binary operation [ilmath]*[/ilmath] forms a group[2]

  • Identity element
  • Inverses
  • Association

See Homotopy class for these properties

TODO: Mond p30


See also

References

  1. ↑ Introduction to Topology - Second Edition - Theodore W. Gamelin and Rober Everist Greene
  2. ↑ Introduction to topology - lecture notes nov 2013 - David Mond


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