Difference between revisions of "HomotopyPage"
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| + | : '''Note: ''' a homotopy is really a relation on continuous functions (see [[homotopic]]), however since any continuous map of the form {{M|F:X\times I\rightarrow Y}} is a homotopy, and has continuous initial and final stages, it also means that, automatically, it's initial and final stages are [[homotopic maps]]. So simply by exhibiting a continuous function, {{M|F:X\times I\rightarrow Y}}, we get homotopic maps. So a homotopy warrants a definition, even if it is useless by itself. | ||
| + | __TOC__ | ||
==Definition== | ==Definition== | ||
Let {{Top.|X|J}} and {{Top.|Y|K}} be {{plural|topological space|s}} and let {{M|A\in\mathcal{P}(X)}} be an arbitrary subset of {{M|X}}. A ''homotopy, relative to {{M|A}}'' is, in its purest form, is any ''[[continuous]]'' [[map]]: | Let {{Top.|X|J}} and {{Top.|Y|K}} be {{plural|topological space|s}} and let {{M|A\in\mathcal{P}(X)}} be an arbitrary subset of {{M|X}}. A ''homotopy, relative to {{M|A}}'' is, in its purest form, is any ''[[continuous]]'' [[map]]: | ||
| − | * {{M|F:X\times I\rightarrow Y}} (where {{M|1=I:=[0,1]\subset\mathbb{R} }} - the [[unit interval]]) such that {{M|1=\forall a\in A\forall s,t\in I[F(a,t)=F(a,s)]}}<ref group="Note">Note that if {{M|1=A=\emptyset}} then there is no {{M|a\in A}} and this represents no condition</ref> | + | * {{M|F:X\times I\rightarrow Y}} (where {{M|1=I:=[0,1]\subset\mathbb{R} }} - the [[unit interval]]) |
| + | ** such that {{M|1=\forall a\in A\forall s,t\in I[F(a,t)=F(a,s)]}}<ref group="Note">Note that if {{M|1=A=\emptyset}} then there is no {{M|a\in A}} and this represents no condition</ref> - the homotopy is fixed on {{M|A}}. | ||
| + | There is some terminology used depending on whether or not {{M|1=A=\emptyset}}: | ||
| + | # {{M|1=A=\emptyset}} then we call {{M|F}} a ''free homotopy'' or just ''homotopy''. If however | ||
| + | # {{M|1=A\ne\emptyset}} then we speak of a ''homotopy relative to {{M|A}}'' or {{M|F\ (\text{rel}\ A)}} | ||
===Stages of the homotopy=== | ===Stages of the homotopy=== | ||
Let {{M|t\in I}} be given, and {{M|H:X\times I\rightarrow Y}} be a homotopy as defined above. {{M|h_t:X\rightarrow Y}} denotes a ''stage of the homotopy'' and is defined as follows: | Let {{M|t\in I}} be given, and {{M|H:X\times I\rightarrow Y}} be a homotopy as defined above. {{M|h_t:X\rightarrow Y}} denotes a ''stage of the homotopy'' and is defined as follows: | ||
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==References== | ==References== | ||
<references/> | <references/> | ||
| − | {{Definition|Topology}} | + | {{Definition|Topology|Homotopy Theory}} |
Latest revision as of 12:53, 15 September 2016
- Note: a homotopy is really a relation on continuous functions (see homotopic), however since any continuous map of the form F:X×I→Y is a homotopy, and has continuous initial and final stages, it also means that, automatically, it's initial and final stages are homotopic maps. So simply by exhibiting a continuous function, F:X×I→Y, we get homotopic maps. So a homotopy warrants a definition, even if it is useless by itself.
Contents
[hide]Definition
Let (X,J) and (Y,K) be topological spaces and let A∈P(X) be an arbitrary subset of X. A homotopy, relative to A is, in its purest form, is any continuous map:
- F:X×I→Y (where I:=[0,1]⊂R - the unit interval)
- such that ∀a∈A∀s,t∈I[F(a,t)=F(a,s)][Note 1] - the homotopy is fixed on A.
There is some terminology used depending on whether or not A=∅:
- A=∅ then we call F a free homotopy or just homotopy. If however
- A≠∅ then we speak of a homotopy relative to A or F (rel A)
Stages of the homotopy
Let t∈I be given, and H:X×I→Y be a homotopy as defined above. ht:X→Y denotes a stage of the homotopy and is defined as follows:
- ht:x↦H(x,t)
The family of stages, {ht:X→Y}t∈I, are collectively called the stages of the homotopy and
- h0:X→Y defined by h0:x↦H(x,0) is the initial stage of the homotopy.
- h1:X→Y defined by h1:x↦H(x,1) is the final stage of the homotopy.
Note that the stages of a homotopy are continuous
Purpose
The point of a homotopy is to be a relation (in fact an equivalence relation) of (continuous) maps.
Homotopic maps
Notes
- Jump up ↑ Note that if A=∅ then there is no a∈A and this represents no condition
