The fundamental group
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Contents
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
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
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.
- 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]
- By homotopy invariance of loop concatenation we see exactly the desired result
- Since [ilmath]\ell_1,\ell_2,\ell_1'[/ilmath] and [ilmath]\ell_2'[/ilmath] were arbitrary this holds for all.
- Let [ilmath]\ell_1,\ell_2,\ell_1',\ell_2'\in\Omega(X,b)[/ilmath] be given
- 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]
- We obtain [ilmath]\overline{*}:\pi_1(X,b)\times\pi_1(X,b)\rightarrow\pi_1(X,b)[/ilmath] given unambiguously by:
- Thus the group operation is:
- [ilmath][\ell_1]\overline{*}[\ell_2]:=[\ell_1*\ell_2][/ilmath]
- Setup:
- Associativity of the operation [ilmath]\overline{*} [/ilmath]
- Existence of an identity element in [ilmath](\pi_1(X,b),\overline{*})[/ilmath]
- 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]}
- [math]\pi_1(X,x_0)[/math] denotes the set of homotopy classes of loops based at [ilmath]x_0[/ilmath]
- 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]}
See also
References
- ↑ Introduction to Topology - Second Edition - Theodore W. Gamelin and Rober Everist Greene
- ↑ Introduction to topology - lecture notes nov 2013 - David Mond
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