Difference between revisions of "A continuous map induces a homomorphism on fundamental groups"

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m (→‎Formal definition: forgot to hide overflow)
(Using diagram template - added proof, nearly finished)
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{{Stub page|grade=A|msg=Important work!}}
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{{Stub page|grade=A|msg=Important work!
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# The definition is a little messy and due to the clutter doesn't show the most important part:
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#* {{M|\varphi_*([f]):\eq[\varphi\circ f]}}
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}}
 
__TOC__
 
__TOC__
 
==Statement==
 
==Statement==
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Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]], let {{M|\varphi:X\rightarrow Y}} be a [[continuous map]] and let {{M|p\in X}} be the base point we consider for the [[fundamental group]] of {{M|X}} at {{M|p}}, {{M|\pi_1(X,p)}}. Then{{rITTMJML}}:
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Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]], let {{M|\varphi:X\rightarrow Y}} be a [[continuous map]] and let {{M|p\in X}} serve as the base point for the [[fundamental group]] of {{M|X}} at {{M|p}}, {{M|\pi_1(X,p)}}. Then{{rITTMJML}}:
 
* {{M|\varphi_\ast:\pi_1(X,p)\rightarrow\pi_1(Y,\varphi(p))}} defined by {{M|\varphi_\ast:[f]\mapsto[\varphi\circ f]}} is a [[group homomorphism|homomorphism of the fundamental groups of {{M|X}} and {{M|Y}}]]
 
* {{M|\varphi_\ast:\pi_1(X,p)\rightarrow\pi_1(Y,\varphi(p))}} defined by {{M|\varphi_\ast:[f]\mapsto[\varphi\circ f]}} is a [[group homomorphism|homomorphism of the fundamental groups of {{M|X}} and {{M|Y}}]]
 
{{Caveat|We are implicitly claiming it is well defined:}} as we do not have {{M|f}} when we write {{M|[f]}}, to obtain {{M|f}} we must look at the inverse relation of the canonical projection, {{M|\mathbb{P}_X^{-1}([f]) }} in the notation developed next, giving us a set of all things equivalent to {{M|f}} and for any of these {{M|\varphi_\ast}} must yield the same result.
 
{{Caveat|We are implicitly claiming it is well defined:}} as we do not have {{M|f}} when we write {{M|[f]}}, to obtain {{M|f}} we must look at the inverse relation of the canonical projection, {{M|\mathbb{P}_X^{-1}([f]) }} in the notation developed next, giving us a set of all things equivalent to {{M|f}} and for any of these {{M|\varphi_\ast}} must yield the same result.
 
* {{M|\varphi_\ast}} is called the [[homomorphism induced by a continuous map|homomorphism induced by the continuous map {{M|\varphi}}]]
 
* {{M|\varphi_\ast}} is called the [[homomorphism induced by a continuous map|homomorphism induced by the continuous map {{M|\varphi}}]]
 
===Formal definition===
 
===Formal definition===
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<div style="float:right;margin:0px;margin-left:0.2em;overflow:hidden;">
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{{Diagram|1=<m>\xymatrix{
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{| class="wikitable" border="1" style="margin:0px;max-width:35em;"
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|-
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| <center><span style="font-size:14pt;"><m>\xymatrix{
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\Omega(X,p) \ar[rr]^-{M_\varphi} \ar[d]_{\mathbb{P}_X} & & \Omega(Y,\varphi(p)) \ar[d]^{\mathbb{P}_Y} \\
 
\Omega(X,p) \ar[rr]^-{M_\varphi} \ar[d]_{\mathbb{P}_X} & & \Omega(Y,\varphi(p)) \ar[d]^{\mathbb{P}_Y} \\
 
\pi_1(X,p) \ar@{.>}[rr]_-{\varphi_*:\eq\overline{M_\varphi} } & & \pi_1(Y,\varphi(p))
 
\pi_1(X,p) \ar@{.>}[rr]_-{\varphi_*:\eq\overline{M_\varphi} } & & \pi_1(Y,\varphi(p))
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}</m></span></center>
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}</m>
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|-
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|2=test}}
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! Diagram
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|}
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</div>
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With our situation we automatically have the following (which do not use their conventional symbols):
 
With our situation we automatically have the following (which do not use their conventional symbols):
 
* {{M|\mathbb{P}_X:}}[[Omega(X,b)|{{M|\Omega(X,p)}}]]{{M|\rightarrow\pi_1(X,p)}}<ref group="Note">{{M|\pi_X}} is not used for the canonical projection because {{M|\pi}} is already in play as the fundamental group. Although it wouldn't lead to ambiguous writings, it's not helpful</ref><ref group="Note">Recall that [[Omega(X,b)|{{M|\Omega(X,p)}}]] is the [[set]] of all {{link|loop|topology|s}} in {{M|X}} based at {{M|p\in X}}. There is an operation, [[loop concatenation]], but it isn't a [[monoid]] or even a [[semigroup]] yet! As concatenation is not associative</ref> is the [[canonical projection of the equivalence relation]]
 
* {{M|\mathbb{P}_X:}}[[Omega(X,b)|{{M|\Omega(X,p)}}]]{{M|\rightarrow\pi_1(X,p)}}<ref group="Note">{{M|\pi_X}} is not used for the canonical projection because {{M|\pi}} is already in play as the fundamental group. Although it wouldn't lead to ambiguous writings, it's not helpful</ref><ref group="Note">Recall that [[Omega(X,b)|{{M|\Omega(X,p)}}]] is the [[set]] of all {{link|loop|topology|s}} in {{M|X}} based at {{M|p\in X}}. There is an operation, [[loop concatenation]], but it isn't a [[monoid]] or even a [[semigroup]] yet! As concatenation is not associative</ref> is the [[canonical projection of the equivalence relation]]
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* {{M|\varphi_\ast(\alpha):\eq\mathbb{P}_Y(M_\varphi(\mathbb{P}^{-1}_X(\alpha)))}} is an unambiguous (i.e. is [[well-defined]]) definition and is a [[group homomorphism]].  
 
* {{M|\varphi_\ast(\alpha):\eq\mathbb{P}_Y(M_\varphi(\mathbb{P}^{-1}_X(\alpha)))}} is an unambiguous (i.e. is [[well-defined]]) definition and is a [[group homomorphism]].  
 
** It is called the [[homomorphism induced by a continuous map|homomorphism induced by the continuous map {{M|\varphi}}]]
 
** It is called the [[homomorphism induced by a continuous map|homomorphism induced by the continuous map {{M|\varphi}}]]
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==Proof==
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===[[Well-definedness]] of {{M|\varphi_*}}===
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{{Diagram|1=<m>\xymatrix{
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\Omega(X,p) \ar[rr]^-{M_\varphi} \ar[d]_{\mathbb{P}_X} & & \Omega(Y,\varphi(p)) \ar[d]^{\mathbb{P}_Y} \\
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\pi_1(X,p) \ar@{.>}[rr]_-{\varphi_*:\eq\overline{M_\varphi} } & & \pi_1(Y,\varphi(p))
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}</m>
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|2=test}}
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We wish to {{link|factor|function}} to yield the map {{M|\overline{M_\varphi} }} as shown on the right. To apply the theorem we must first show:
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* {{M|\forall f,g\in\Omega(X,p)\big[\big(\mathbb{P}_X(f)\eq\mathbb{P}_X(g)\big)\implies\big(\mathbb{P}_Y(M_\varphi(f))\eq\mathbb{P}_Y(M_\varphi(g))\big)\big]}}
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'''Proof:'''
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* Let {{M|f,g\in\Omega(X,p)}} be given
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** Suppose that {{M|\mathbb{P}_X(f)\neq\mathbb{P}_X(g)}} - then by the nature of [[logical implication]] we're done, we do not care about the truth or falsity of the RHS
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** Suppose that {{M|\mathbb{P}_X(f)\eq\mathbb{P}_X(g)}} - then we must show that in this case {{M|\mathbb{P}_Y(M_\varphi(f))\eq\mathbb{P}_Y(M_\varphi(g))}}
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*** We have {{M|\mathbb{P}_X(f)\eq\mathbb{P}_X(g)}}, that means:
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**** {{M|f\simeq g\ (\text{rel }\{0,1\})}}
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***** By [[the composition of end-point-preserving-homotopic paths with a continuous map yields end-point-preserving-homotopic paths]], this means:
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****** {{M|(\varphi\circ f)\simeq(\varphi\circ g)\ (\text{rel }\{0,1\})}}
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*** Thus {{M|\mathbb{P}_X(f)\eq\mathbb{P}_X(g)\implies (\varphi\circ f)\simeq(\varphi\circ g)\ (\text{rel }\{0,1\})}}
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**** Furthermore {{M|(\varphi\circ f)\simeq(\varphi\circ g)\ (\text{rel }\{0,1\})}} is exactly the definition of {{M|[\varphi\circ f]\eq[\varphi\circ g]}} (i.e. {{M|\mathbb{P}_Y(\varphi\circ f)\eq\mathbb{P}_Y(\varphi\circ g)}}) (which are obviously in {{M|\pi_1(Y,\varphi(p))}} of course)
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*** Next let us simplify the RHS:
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***# {{M|\mathbb{P}_Y(M_\varphi(f))}}
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***#: {{M|\eq\mathbb{P}_Y(\varphi\circ f)}}
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***#: {{M|\eq[\varphi\circ f]}}
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***# {{M|\mathbb{P}_Y(M_\varphi(g))}}
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***#: {{M|\eq\mathbb{P}_Y(\varphi\circ g)}}
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***#: {{M|\eq[\varphi\circ g]}}
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*** We have already shown that we have {{M|[\varphi\circ f]\eq[\varphi\circ g]}} from the LHS
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** So the entire thing boiled down to:
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*** [[the composition of end-point-preserving-homotopic paths with a continuous map yields end-point-preserving-homotopic paths|{{M|\big[f\simeq g\ (\text{rel }\{0,1\})\big]\implies\big[(\varphi\circ f)\simeq(\varphi\circ g)\ (\text{rel }\{0,1\})\big]}}]]
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* Since {{M|f,g\in\Omega(X,p)}} we arbitrary we have shown it for all.
  
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==OLD Statement==
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Thus we may define {{M|\overline{M_\varphi}:\pi_1(X,p)\rightarrow\pi_1(Y,\varphi(p))}} unambiguously by {{M|\overline{M_\varphi}:\alpha\mapsto\mathbb{P}_Y(M_\varphi(\mathbb{P}^{-1}_X(\alpha)))}} - it doesn't matter which representative of {{M|\mathbb{P}^{-1}_X(\alpha)}} we take.
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Let {{Top.|X|J}} and {{Top.|Y|K}} be [[topological spaces]], let {{M|\varphi:X\rightarrow Y}} be a [[continuous map]] and let {{M|p\in X}} be the base point we consider for the [[fundamental group]] of {{M|X}} at {{M|p}}, {{M|\pi_1(X,p)}} then we also have the following:
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* {{M|\mathbb{P}_X:}}[[Omega(X,b)|{{M|\Omega(X,p)}}]]{{M|\rightarrow\pi_1(X,p)}}<ref group="Note">{{M|\pi_X}} is not used for the canonical projection because {{M|\pi}} is already in play as the fundamental group. Although it wouldn't lead to ambiguous writings, it's not helpful</ref><ref group="Note">Recall that [[Omega(X,b)|{{M|\Omega(X,p)}}]] is the [[set]] of all {{link|loop|topology|s}} in {{M|X}} based at {{M|p\in X}}. There is an operation, [[loop concatenation]], but it isn't a [[monoid]] or even a [[semigroup]] yet! As concatenation is not associative</ref> is the [[canonical projection of the equivalence relation]], i.e. {{M|\mathbb{P}_X:f\mapsto [f]\in\frac{\Omega(X,p)}{\big({\small(\cdot)}\simeq{\small(\cdot)}\ (\text{rel }\{0,1\})\big)} }};
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* {{M|\mathbb{P}_Y:\Omega(Y,\varphi(p))\rightarrow\pi_1(Y,\varphi(p))}} is the canonical projection as above but for {{M|Y}}, and
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* {{M|M:\Omega(X,p)\rightarrow\Omega(Y,\varphi(p))}} by {{M|M:f\mapsto(\varphi\circ f)}}
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In this case we claim that{{rITTMJML}}:
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* {{M|\varphi_\ast(\alpha):\eq\mathbb{P}_Y(M(\mathbb{P}^{-1}_X(\alpha)))}} is an unambiguous (i.e. is [[well-defined]]) definition and is a [[group homomorphism]].
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** It is called the [[homomorphism induced by a continuous map|homomorphism induced by the continuous map {{M|\varphi}}]]
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===Informal definition===
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Informally, we define {{M|\varphi_\ast}} as follows:
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* {{M|\varphi_\ast:\pi_1(X,p)\rightarrow\pi_1(Y,\varphi(p))}} defined by {{M|\varphi_\ast:[f]\mapsto[\varphi\circ f]}} is a [[group homomorphism]]
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We claim that this is [[well-defined]] and that it is indeed a [[group homomorphism]]
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This justifies the notation {{M|\overline{M_\varphi}:[f]\mapsto [\varphi\circ f]}} - as it doesn't matter whuch {{M|f}} we take to represent the [[equivalence class] {{M|[f]}}.
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==Proof==
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===[[Well-definedness]] of {{M|\varphi_*}}===
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The notation {{M|\varphi_*([f])}} makes it seem like {{M|f}} is some given loop. Remember that we're actually dealing with [[equivalence classes]] not a loop, thus:
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* for {{M|\alpha\in\pi_1(X,p)}} we must define {{M|\varphi_*(\alpha)}} - not so obvious now! We actually define:
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** {{M|\varphi_*(\alpha):\eq [f\circ\mathbb{P}^{-1}_X
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===Group homomorphism===
 
===Group homomorphism===
 
We want to show that:
 
We want to show that:
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** Clearly these are the same
 
** Clearly these are the same
 
* Since {{M|[f],[g]\in\pi_1(X,p)}} were arbitrary we have shown this for all. As required.
 
* Since {{M|[f],[g]\in\pi_1(X,p)}} were arbitrary we have shown this for all. As required.
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===Stuff===
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{{Proofreading of proof required|grade=D|msg=Typos at most}}
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By [[the composition of end-point-preserving-homotopic paths with a continuous map yields end-point-preserving-homotopic paths]] we know that if:
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* {{M|f_1,f_2:I\rightarrow X}} are {{link|path|topology|s}} and {{M|\varphi}} is a continuous map, as stated above, that:
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** If {{M|f_1\simeq f_2\ (\text{rel }\{0,1\})}} then {{M|(\varphi\circ f_1)\simeq(\varphi\circ f_2)\ (\text{rel }\{0,1\})}}
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==Notes==
 
==Notes==
 
<references group="Note"/>
 
<references group="Note"/>

Revision as of 07:12, 13 December 2016

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This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Important work!
  1. The definition is a little messy and due to the clutter doesn't show the most important part:
    • [ilmath]\varphi_*([f]):\eq[\varphi\circ f][/ilmath]

Statement

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces, let [ilmath]\varphi:X\rightarrow Y[/ilmath] be a continuous map and let [ilmath]p\in X[/ilmath] serve as the base point for the fundamental group of [ilmath]X[/ilmath] at [ilmath]p[/ilmath], [ilmath]\pi_1(X,p)[/ilmath]. Then[1]:

Caveat:We are implicitly claiming it is well defined: as we do not have [ilmath]f[/ilmath] when we write [ilmath][f][/ilmath], to obtain [ilmath]f[/ilmath] we must look at the inverse relation of the canonical projection, [ilmath]\mathbb{P}_X^{-1}([f]) [/ilmath] in the notation developed next, giving us a set of all things equivalent to [ilmath]f[/ilmath] and for any of these [ilmath]\varphi_\ast[/ilmath] must yield the same result.

Formal definition

[ilmath]\xymatrix{ \Omega(X,p) \ar[rr]^-{M_\varphi} \ar[d]_{\mathbb{P}_X} & & \Omega(Y,\varphi(p)) \ar[d]^{\mathbb{P}_Y} \\ \pi_1(X,p) \ar@{.>}[rr]_-{\varphi_*:\eq\overline{M_\varphi} } & & \pi_1(Y,\varphi(p)) }[/ilmath]
test

With our situation we automatically have the following (which do not use their conventional symbols):

  • [ilmath]\mathbb{P}_X:[/ilmath][ilmath]\Omega(X,p)[/ilmath][ilmath]\rightarrow\pi_1(X,p)[/ilmath][Note 1][Note 2] is the canonical projection of the equivalence relation
    • i.e. [ilmath]\mathbb{P}_X:f\mapsto [f]\in\frac{\Omega(X,p)}{\big({\small(\cdot)}\simeq{\small(\cdot)}\ (\text{rel }\{0,1\})\big)} [/ilmath];
  • [ilmath]\mathbb{P}_Y:\Omega(Y,\varphi(p))\rightarrow\pi_1(Y,\varphi(p))[/ilmath] is the canonical projection as above but for [ilmath]Y[/ilmath], and
  • [ilmath]M_\varphi:\Omega(X,p)\rightarrow\Omega(Y,\varphi(p))[/ilmath] by [ilmath]M:f\mapsto(\varphi\circ f)[/ilmath] is the core of the definition, the map taking loops to their images

In this case we claim that[1]:

Proof

Well-definedness of [ilmath]\varphi_*[/ilmath]

[ilmath]\xymatrix{ \Omega(X,p) \ar[rr]^-{M_\varphi} \ar[d]_{\mathbb{P}_X} & & \Omega(Y,\varphi(p)) \ar[d]^{\mathbb{P}_Y} \\ \pi_1(X,p) \ar@{.>}[rr]_-{\varphi_*:\eq\overline{M_\varphi} } & & \pi_1(Y,\varphi(p)) }[/ilmath]
test

We wish to factor to yield the map [ilmath]\overline{M_\varphi} [/ilmath] as shown on the right. To apply the theorem we must first show:

  • [ilmath]\forall f,g\in\Omega(X,p)\big[\big(\mathbb{P}_X(f)\eq\mathbb{P}_X(g)\big)\implies\big(\mathbb{P}_Y(M_\varphi(f))\eq\mathbb{P}_Y(M_\varphi(g))\big)\big][/ilmath]

Proof:

  • Let [ilmath]f,g\in\Omega(X,p)[/ilmath] be given
    • Suppose that [ilmath]\mathbb{P}_X(f)\neq\mathbb{P}_X(g)[/ilmath] - then by the nature of logical implication we're done, we do not care about the truth or falsity of the RHS
    • Suppose that [ilmath]\mathbb{P}_X(f)\eq\mathbb{P}_X(g)[/ilmath] - then we must show that in this case [ilmath]\mathbb{P}_Y(M_\varphi(f))\eq\mathbb{P}_Y(M_\varphi(g))[/ilmath]
      • We have [ilmath]\mathbb{P}_X(f)\eq\mathbb{P}_X(g)[/ilmath], that means:
      • Thus [ilmath]\mathbb{P}_X(f)\eq\mathbb{P}_X(g)\implies (\varphi\circ f)\simeq(\varphi\circ g)\ (\text{rel }\{0,1\})[/ilmath]
        • Furthermore [ilmath](\varphi\circ f)\simeq(\varphi\circ g)\ (\text{rel }\{0,1\})[/ilmath] is exactly the definition of [ilmath][\varphi\circ f]\eq[\varphi\circ g][/ilmath] (i.e. [ilmath]\mathbb{P}_Y(\varphi\circ f)\eq\mathbb{P}_Y(\varphi\circ g)[/ilmath]) (which are obviously in [ilmath]\pi_1(Y,\varphi(p))[/ilmath] of course)
      • Next let us simplify the RHS:
        1. [ilmath]\mathbb{P}_Y(M_\varphi(f))[/ilmath]
          [ilmath]\eq\mathbb{P}_Y(\varphi\circ f)[/ilmath]
          [ilmath]\eq[\varphi\circ f][/ilmath]
        2. [ilmath]\mathbb{P}_Y(M_\varphi(g))[/ilmath]
          [ilmath]\eq\mathbb{P}_Y(\varphi\circ g)[/ilmath]
          [ilmath]\eq[\varphi\circ g][/ilmath]
      • We have already shown that we have [ilmath][\varphi\circ f]\eq[\varphi\circ g][/ilmath] from the LHS
    • So the entire thing boiled down to:
  • Since [ilmath]f,g\in\Omega(X,p)[/ilmath] we arbitrary we have shown it for all.

Thus we may define [ilmath]\overline{M_\varphi}:\pi_1(X,p)\rightarrow\pi_1(Y,\varphi(p))[/ilmath] unambiguously by [ilmath]\overline{M_\varphi}:\alpha\mapsto\mathbb{P}_Y(M_\varphi(\mathbb{P}^{-1}_X(\alpha)))[/ilmath] - it doesn't matter which representative of [ilmath]\mathbb{P}^{-1}_X(\alpha)[/ilmath] we take.

This justifies the notation [ilmath]\overline{M_\varphi}:[f]\mapsto [\varphi\circ f][/ilmath] - as it doesn't matter whuch [ilmath]f[/ilmath] we take to represent the [[equivalence class] [ilmath][f][/ilmath].

Group homomorphism

We want to show that:

  • [ilmath]\forall [f],[g]\in\pi_1(X,p)\big[\varphi_\ast([f]\cdot[g])\eq\varphi_\ast([f])\cdot\varphi_\ast([g])\big][/ilmath]

We will do this by operating on the left-hand-side (LHS) and the right-hand-side (RHS) separately.

  • Let [ilmath][f],[g]\in\pi_1(X,p)[/ilmath] be given.
    • We now operate on the LHS and RHS:
      1. The LHS:
        • [ilmath]\varphi_\ast([f]\cdot[g])[/ilmath]
          [ilmath]\eq\varphi_\ast([f*g])[/ilmath] (by the operation of the fundamental group) - note that [ilmath]*[/ilmath] here denotes loop concatenation of course.
          [ilmath]\eq[\varphi\circ(f*g)][/ilmath] (by definition of [ilmath]\varphi_\ast[/ilmath])
      2. The RHS:
        • [ilmath]\varphi_\ast([f])\cdot\varphi_\ast([g])[/ilmath]
          [ilmath]\eq[\varphi\circ f]\cdot[\varphi\circ g][/ilmath]
          [ilmath]\eq[(\varphi\circ f)*(\varphi\circ g)][/ilmath]
    • Now we must show they're equal.
      1. Using the definition of loop concatenation we see [ilmath]\text{LHS}\eq\varphi\circ\left(\left\{\begin{array}{lr}f(2t)&\text{for }t\in[0,\frac{1}{2}]\\ g(2t-1)&\text{for }t\in[\frac{1}{2},1]\end{array}\right.\right)[/ilmath][ilmath]\eq\left\{\begin{array}{lr}\varphi(f(2t))&\text{for }t\in[0,\frac{1}{2}]\\\varphi(g(2t-1)&\text{for }t\in[\frac{1}{2},1]\end{array}\right.[/ilmath]
      2. Also using the definition of loop concatenation we see [ilmath]\text{RHS}\eq\left\{\begin{array}{lr}\varphi(f(2t))&\text{for }t\in[0,\frac{1}{2}]\\\varphi(g(25-1))&\text{for }t\in[\frac{1}{2},1]\end{array}\right.[/ilmath]
    • Clearly these are the same
  • Since [ilmath][f],[g]\in\pi_1(X,p)[/ilmath] were arbitrary we have shown this for all. As required.

Template:Proofreading of proof required

Notes

  1. ↑ [ilmath]\pi_X[/ilmath] is not used for the canonical projection because [ilmath]\pi[/ilmath] is already in play as the fundamental group. Although it wouldn't lead to ambiguous writings, it's not helpful
  2. ↑ Recall that [ilmath]\Omega(X,p)[/ilmath] is the set of all loops in [ilmath]X[/ilmath] based at [ilmath]p\in X[/ilmath]. There is an operation, loop concatenation, but it isn't a monoid or even a semigroup yet! As concatenation is not associative

References

  1. ↑ 1.0 1.1 Introduction to Topological Manifolds - John M. Lee


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