Difference between revisions of "Homotopy"
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==Definition== | ==Definition== | ||
Given two [[topological spaces]], {{Top.|X|J}} and {{Top.|Y|K}} then a ''homotopy of maps (from {{M|X}} to {{M|Y}})'' is a ''[[continuous]]'' [[function]]: {{M|F:X\times I\rightarrow Y}} (where {{M|I}} denotes the [[unit interval]], {{M|1=I:=[0,1]\subset\mathbb{R} }}). Note: | Given two [[topological spaces]], {{Top.|X|J}} and {{Top.|Y|K}} then a ''homotopy of maps (from {{M|X}} to {{M|Y}})'' is a ''[[continuous]]'' [[function]]: {{M|F:X\times I\rightarrow Y}} (where {{M|I}} denotes the [[unit interval]], {{M|1=I:=[0,1]\subset\mathbb{R} }}). Note: | ||
| − | * The ''stages of the homotopy, {{M|F}},'' are a family of functions, {{M|\{ f_t:X\rightarrow Y\ \vert\ t\in[0,1]\} }} such that {{M|f_t:x\rightarrow F(x,t)}}. [[The stages of a homotopy are continuous]]. | + | * The '''''stages of the homotopy, {{M|F}},''''' are a family of functions, {{M|\{ f_t:X\rightarrow Y\ \vert\ t\in[0,1]\} }} such that {{M|f_t:x\rightarrow F(x,t)}}. [[The stages of a homotopy are continuous]]. |
** {{M|f_0}} and {{M|f_1}} are examples of stages, and are often called the ''initial stage of the homotopy'' and ''final stage of the homotopy'' respectively. | ** {{M|f_0}} and {{M|f_1}} are examples of stages, and are often called the ''initial stage of the homotopy'' and ''final stage of the homotopy'' respectively. | ||
Two ([[continuous]]) functions, {{M|g,h:X\rightarrow Y}} are said to be ''homotopic'' if there exists a homotopy such that {{M|1=f_0=g}} and {{M|1=f_1=h}} | Two ([[continuous]]) functions, {{M|g,h:X\rightarrow Y}} are said to be ''homotopic'' if there exists a homotopy such that {{M|1=f_0=g}} and {{M|1=f_1=h}} | ||
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Contents
Definition
Given two topological spaces, [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] then a homotopy of maps (from [ilmath]X[/ilmath] to [ilmath]Y[/ilmath]) is a continuous function: [ilmath]F:X\times I\rightarrow Y[/ilmath] (where [ilmath]I[/ilmath] denotes the unit interval, [ilmath]I:=[0,1]\subset\mathbb{R}[/ilmath]). Note:
- The stages of the homotopy, [ilmath]F[/ilmath], are a family of functions, [ilmath]\{ f_t:X\rightarrow Y\ \vert\ t\in[0,1]\} [/ilmath] such that [ilmath]f_t:x\rightarrow F(x,t)[/ilmath]. The stages of a homotopy are continuous.
- [ilmath]f_0[/ilmath] and [ilmath]f_1[/ilmath] are examples of stages, and are often called the initial stage of the homotopy and final stage of the homotopy respectively.
Two (continuous) functions, [ilmath]g,h:X\rightarrow Y[/ilmath] are said to be homotopic if there exists a homotopy such that [ilmath]f_0=g[/ilmath] and [ilmath]f_1=h[/ilmath]
Notes
- ↑ Do not shorten this to "homotopy equivalence" as homotopy equivalence of spaces is something very different
References
Template:Algebraic topology navbox
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