Difference between revisions of "Homotopy class"
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* Symmetric: {{M|\alpha\simeq\beta\text{ rel}\{0,1\}\implies \beta\simeq\alpha\text{ rel}\{0,1\} }} | * Symmetric: {{M|\alpha\simeq\beta\text{ rel}\{0,1\}\implies \beta\simeq\alpha\text{ rel}\{0,1\} }} | ||
* Transitive: {{M|\alpha\simeq\beta\text{ rel}\{0,1\}\wedge\beta\simeq\gamma\text{ rel}\{0,1\}\implies \alpha\simeq\gamma\text{ rel}\{0,1\} }} | * Transitive: {{M|\alpha\simeq\beta\text{ rel}\{0,1\}\wedge\beta\simeq\gamma\text{ rel}\{0,1\}\implies \alpha\simeq\gamma\text{ rel}\{0,1\} }} | ||
| + | |||
| + | The [[Equivalence class|equivalence class]] of {{M|\alpha}} is denoted (as is usual) by {{M|[\alpha]}} | ||
| + | ==Important properties== | ||
| + | {{M|\alpha,\ \beta}} and {{M|\gamma}} denote paths | ||
| + | |||
| + | |||
| + | * For a [[Continuous map|continuous map]] {{M|p:[0,1]\rightarrow[0,1]}} with {{M|1=p(0)=0}} and {{M|1=p(1)=1}} we have: | ||
| + | *: {{M|1=[\alpha\circ p]=[\alpha]}} - that is ''any [[Reparametrisation|reparametrisation]] of {{M|\alpha}} is [[Homotopic paths|homotopic]] to {{M|\alpha}}'' | ||
| + | * <math>[\alpha_1]=[\alpha_2]\wedge[\beta_1]=[\beta_2]\implies[\alpha_1\beta_1]=[\alpha_2\beta_2]</math> | ||
| + | ** This allows us to define multiplication | ||
| + | * <math>(\alpha\beta)\gamma\simeq\alpha(\beta\gamma)\text{ rel}\{0,1\}</math> or <math>([\alpha][\beta])[\gamma]=[\alpha]([\beta][\gamma])</math> | ||
| + | ** This allows us to define associativity | ||
| + | * Where {{M|a}} is the constant loop at {{M|a}} (ie {{M|1=a(t)=a\ \forall t\in[0,1]}}) we have | ||
| + | *: <math>a\alpha\simeq\alpha\simeq\alpha b\text{ rel}\{0,1\}</math> or <math>[a][\alpha]=[\alpha]=[\alpha][b]</math> | ||
| + | * if {{M|\alpha^{-1} }} is the reverse path of {{M|\alpha}}, literally {{M|1=\alpha^{-1}(t)=\alpha(1-t)}} then | ||
| + | ** {{M|1=[\alpha_0]=[\alpha_1]\implies[\alpha_0^{-1}]=[\alpha_1^{-1}]}} | ||
| + | *** we can now define the inverse, {{M|1=[\alpha^{-1}]=[\alpha]^{-1} }} | ||
| + | * {{M|1=[\alpha][\alpha]^{-1}=[a]}} | ||
| + | |||
| + | {{Todo|Proofs for all of these p117}} | ||
==See also== | ==See also== | ||
Latest revision as of 02:37, 17 April 2015
Definition
The relation of paths being end-point-preserving homotopic is an Equivalence relation[1]
That is [ilmath]\alpha\simeq\beta\text{ rel}\{0,1\} [/ilmath] where [ilmath]\alpha[/ilmath] and [ilmath]\beta[/ilmath] are paths from [ilmath]a[/ilmath] to [ilmath]b[/ilmath] (which are not necessarily distinct as it may be a loop) is an equivalence relation, which is to say:
- Reflexive: [ilmath]\alpha\simeq\alpha\text{ rel}\{0,1\} [/ilmath]
- Symmetric: [ilmath]\alpha\simeq\beta\text{ rel}\{0,1\}\implies \beta\simeq\alpha\text{ rel}\{0,1\} [/ilmath]
- Transitive: [ilmath]\alpha\simeq\beta\text{ rel}\{0,1\}\wedge\beta\simeq\gamma\text{ rel}\{0,1\}\implies \alpha\simeq\gamma\text{ rel}\{0,1\} [/ilmath]
The equivalence class of [ilmath]\alpha[/ilmath] is denoted (as is usual) by [ilmath][\alpha][/ilmath]
Important properties
[ilmath]\alpha,\ \beta[/ilmath] and [ilmath]\gamma[/ilmath] denote paths
- For a continuous map [ilmath]p:[0,1]\rightarrow[0,1][/ilmath] with [ilmath]p(0)=0[/ilmath] and [ilmath]p(1)=1[/ilmath] we have:
- [ilmath][\alpha\circ p]=[\alpha][/ilmath] - that is any reparametrisation of [ilmath]\alpha[/ilmath] is homotopic to [ilmath]\alpha[/ilmath]
- [math][\alpha_1]=[\alpha_2]\wedge[\beta_1]=[\beta_2]\implies[\alpha_1\beta_1]=[\alpha_2\beta_2][/math]
- This allows us to define multiplication
- [math](\alpha\beta)\gamma\simeq\alpha(\beta\gamma)\text{ rel}\{0,1\}[/math] or [math]([\alpha][\beta])[\gamma]=[\alpha]([\beta][\gamma])[/math]
- This allows us to define associativity
- Where [ilmath]a[/ilmath] is the constant loop at [ilmath]a[/ilmath] (ie [ilmath]a(t)=a\ \forall t\in[0,1][/ilmath]) we have
- [math]a\alpha\simeq\alpha\simeq\alpha b\text{ rel}\{0,1\}[/math] or [math][a][\alpha]=[\alpha]=[\alpha][b][/math]
- if [ilmath]\alpha^{-1} [/ilmath] is the reverse path of [ilmath]\alpha[/ilmath], literally [ilmath]\alpha^{-1}(t)=\alpha(1-t)[/ilmath] then
- [ilmath][\alpha_0]=[\alpha_1]\implies[\alpha_0^{-1}]=[\alpha_1^{-1}][/ilmath]
- we can now define the inverse, [ilmath][\alpha^{-1}]=[\alpha]^{-1}[/ilmath]
- [ilmath][\alpha_0]=[\alpha_1]\implies[\alpha_0^{-1}]=[\alpha_1^{-1}][/ilmath]
- [ilmath][\alpha][\alpha]^{-1}=[a][/ilmath]
TODO: Proofs for all of these p117
See also
References
- ↑ Introduction to topology - Second Edition - Theodore W. Gamelin and Rober Everist Greene
