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I cannot believe it's been 15 months and this still isn't complete!

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

Definition

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

π1(X,b):=Loop(X,b)((⋅)≃(⋅) (rel {0,1})) has a group structure, with the group operation being:
- $:[\ell_1]\cdot[\ell_2]\mapsto[\ell_1*\ell_2]$ where $\ell_1*\ell_2$ denotes the loop concatenation of $\ell_1,\ell_2\in\text{Loop}(X,b)$ .

Proof of claims

[Expand]

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

Factoring $*$ (loop concatenation) setup
$\xymatrix{ \Omega(X,b)\times\Omega(X,b) \ar@2{->}[d]_{(\pi,\pi)} \ar[rr]^-{} \ar@/^3.5ex/[drr]^(.75){\pi\circ } & & \Omega(X,b) \ar[d]^{\pi} \\ \pi_1(X,b)\times\pi_1(X,b) \ar@{.>}[rr]_-{\overline{*} } & & \pi_1(X,b) }$

Let

$(X,\mathcal{ J })$ be a topological space and let

$b\in X$ be given. Then

$\Omega(X,b)$ is the set of all loops based at

$b$ . Let

${\small(\cdot)}\simeq{\small(\cdot)}\ (\text{rel }\{0,1\})$ denote the relation of end-point-preserving homotopy on

$C([0,1],X)$ - the set of all paths in

$X$ - but considered only on the subset of

$C([0,1],X)$ ,

$\Omega(X,b)$ .

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

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

∗:(ℓ1,ℓ2)↦(ℓ1∗ℓ2:[0,1]→X given by ℓ1∗ℓ2:t↦{ℓ1(2t)for t∈[0,12]ℓ2(2t−1)for t∈[12,1])
- notice that $t=\frac{1}{2}$ is in both parts, this is a nod to the pasting lemma

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

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

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

∀(ℓ1,ℓ2),(ℓ′1,ℓ′2)∈Ω(X,b)×Ω(X,b)[((π,π)(ℓ1,ℓ2)=(π,π)(ℓ′1,ℓ′2))⟹(π(ℓ1∗ℓ2)=π(ℓ′1∗ℓ′2))], this can be written better using our standard notation:
- $\forall\ell_1,\ell_2,\ell_1',\ell_2'\in\Omega(X,b)\big[\big(([\ell_1],[\ell_2])=([\ell_1'],[\ell_2'])\big)\implies\big([\ell_1*\ell_2]=[\ell_1'*\ell_2']\big)\big]$

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

¯∗:π1(X,b)×π1(X,b)→π1(X,b) given (unambiguously) by ¯∗:([ℓ1],[ℓ2])↦[ℓ1∗ℓ2] or written more nicely as:
- $[\ell_1]\overline{*}[\ell_2]:=[\ell_1*\ell_2]$

Lastly we show $(\pi_1(X,b),\overline{*})$ forms a group

[Expand]

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

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

Factoring:
- Setup:
  - $*:\Omega(X,b)\times\Omega(X,b)\rightarrow\Omega(X,b)$ - the operation of loop concatenation - $*:(\ell_1,\ell_2)\mapsto\left(\ell_1*\ell_2:I\rightarrow X\text{ by }\ell_1*\ell_2:t\mapsto\left\{\begin{array}{lr}\ell_1(2t) & \text{for }t\in[0,\frac{1}{2}]\\ \ell(2t-1) & \text{for }t\in[\frac{1}{2},1]\end{array}\right.\right)$
  through
  - (p,p):Ω(X,b)×Ω(X,b)→π1(X,b)×π1(X,b) by (p,p):(ℓ1,ℓ2)↦(p(ℓ1),p(ℓ2))
    - where $p:\Omega(X,b)\rightarrow\pi_1(X,b)$ is the canonical projection of the equivalence relation. As such we may say:
    - $(p,p)$ is given by by $(p,p):(\ell_1,\ell_2)\mapsto([\ell_1],[\ell_2])$ instead
  - We must show:
    - $\forall\ell_1,\ell_2,\ell_1',\ell_2'\in\Omega(X,b)\left[\big([\ell_1]=[\ell_1']\wedge[\ell_2]=[\ell_2']\big)\implies\big([\ell_1*\ell_2]=[\ell_1'*\ell_2']\big)\right]$ ^{[Note 1]}
- Proof:
  - Let ℓ1,ℓ2,ℓ′1,ℓ′2∈Ω(X,b) be given
    - Suppose that $\neg([\ell_1]=[\ell_1']\wedge[\ell_2]=[\ell_2'])$ 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 [ℓ1]=[ℓ′1]∧[ℓ2]=[ℓ′2] holds. We must show that in this case we have [ℓ1∗ℓ2]=[ℓ′1∗ℓ′2]
      - By homotopy invariance of loop concatenation we see exactly the desired result
  - Since $\ell_1,\ell_2,\ell_1'$ and $\ell_2'$ were arbitrary this holds for all.
- Conclusion
  - We obtain ¯∗:π1(X,b)×π1(X,b)→π1(X,b) given unambiguously by:
    - $\overline{*}:([\ell_1],[\ell_2])\mapsto[\ell_1*\ell_2]$
- Thus the group operation is:
  - $[\ell_1]\overline{*}[\ell_2]:=[\ell_1*\ell_2]$
Associativity of the operation $\overline{*}$
Existence of an identity element in $(\pi_1(X,b),\overline{*})$
For each element of $\pi_1(X,b)$ the existence of an inverse element in $(\pi_1(X,b),\overline{*})$

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References

OLD PAGE

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

Definition

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

denotes the set of homotopy classes of loops based at $x_0$

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

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

References

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

Cite error: <ref> tags exist for a group named "Note", but no corresponding <references group="Note"/> tag was found, or a closing </ref> is missing

[2] Jump up ↑ Introduction to Topology - Second Edition - Theodore W. Gamelin and Rober Everist Greene

[3] Jump up ↑ Introduction to topology - lecture notes nov 2013 - David Mond

[Note 1]

[1]

[2]

@@ Line 1: / Line 1: @@
-'''Requires: ''' [[Paths and loops in a topological space]] and [[Homotopic paths]]<ref>Introduction to topology - lecture notes nov 2013 - David Mond</ref>
+{{Refactor notice|grade=A|msg=I cannot believe it's been 15 months and this still isn't complete!
+* Started refactoring [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 19:55, 1 November 2016 (UTC)}}
 ==Definition==
-Given a [[Topological space|topological space]] {{M|X}} and a point {{M|x_0\in X}}
+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:
-{{Todo|Fundamental group}}
+* {{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:
+** {{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)}}.
+==Proof of claims==
+{{Begin Inline Theorem}}
+[[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]]
+{{Begin Inline Proof}}{{:Proof that the fundamental group is actually a group/Outline}}
+{{End Proof}}{{End Theorem}}
+{{Begin Inline Theorem}}
+[[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]]
+{{Begin Inline Proof}}{{:Proof that the fundamental group is actually a group/Proof}}
+{{End Proof}}{{End Theorem}}
+==References==
+<references/>
+{{Definition|Topology|Homotopy Theory}}
+=OLD PAGE=
+'''Requires: ''' [[Paths and loops in a topological space]] and [[Homotopic paths]]
+==Definition==
+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>
+* <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}}
+: forms a [[Group|group]] under the operation of multiplication of the homotopy classes.
+{{Begin Theorem}}
+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>
+{{Begin Proof}}
+* Identity element
+* Inverses
+* Association
+See [[Homotopy class]] for these properties
+{{Todo|Mond p30}}
+{{End Proof}}
+{{End Theorem}}
+==See also==
+* [[Homotopy class]]
+* [[Homotopic paths]]
+* [[Paths and loops in a topological space]]
 ==References==

Difference between revisions of "The fundamental group"

Latest revision as of 16:10, 4 November 2016

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