Difference between revisions of "Nth homotopy group"
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| + | : '''Note: ''' the [[fundamental group]] is {{M|\pi_1(X,p)}} | ||
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** {{M|f\circ\pi}} is [[continuous]] {{iff}} {{M|f}} is continuous | ** {{M|f\circ\pi}} is [[continuous]] {{iff}} {{M|f}} is continuous | ||
That gives us an association between continuous maps of the form {{M|f\circ\pi}} with some constraints. Blah blah blah, something like that. | That gives us an association between continuous maps of the form {{M|f\circ\pi}} with some constraints. Blah blah blah, something like that. | ||
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| + | [[Pointed topological space|Pointed topological spaces]] are involved. | ||
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==References== | ==References== | ||
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{{Definition|Topology|Algebraic Topology}} | {{Definition|Topology|Algebraic Topology}} | ||
Latest revision as of 22:17, 12 December 2016
- Note: the fundamental group is [ilmath]\pi_1(X,p)[/ilmath]
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Contents
Definition
Let [ilmath](X,\mathcal{J})[/ilmath] be a topological space with [ilmath]x_0\in X[/ilmath] being any fixed point. The [ilmath]n^\text{th} [/ilmath] homotopy group, written [ilmath]\pi_n(X,x_0)[/ilmath] is defined as follows:
- The underlying set of the group is: [math]\pi_n(X,x_0):\eq[(\mathbb{S}^n,p),(X,x_0)]_*[/math]
- Where [ilmath][(\mathbb{S}^n,p),(X,x_0)]_*[/ilmath] denotes equivalence classes of continuous maps where [ilmath]f(p)\eq x_0[/ilmath] under the equivalence relation of homotopy relative to [ilmath]p[/ilmath], i.e.:
- [math][(\mathbb{S}^n,p),(X,x_0)]_*:\eq\frac{\{f\in C(\mathbb{S}^n,X)\ \vert\ f(p)\eq x_0\} }{\big({\small(\cdot)}\ \simeq\ {\small(\cdot)}\ (\text{Rel }\{p\}) \big)} [/math]
- Where [ilmath][(\mathbb{S}^n,p),(X,x_0)]_*[/ilmath] denotes equivalence classes of continuous maps where [ilmath]f(p)\eq x_0[/ilmath] under the equivalence relation of homotopy relative to [ilmath]p[/ilmath], i.e.:
Caveat:I think, the book... pointed spaces are really not that special, I'm using Books:Topology and Geometry - Glen E. Bredon for this
Noting that:
- [ilmath](\mathbb{S}^n,p)\cong\text{RS}\left(\mathbb{S}^{n-1}\right) [/ilmath] where [ilmath]\text{RS} [/ilmath] denotes the reduced suspension of a space we see:
- [math](\mathbb{S}^n,p)\cong\text{RS}\left(\mathbb{S}^{n-1},p\right):\eq\left(\frac{\mathbb{S}^{n-1}\times I}{(\{p\}\times I)\cup(\mathbb{S}^{n-1}\times\{0,1\})},\pi(p,i)\right)[/math] for [ilmath]\pi[/ilmath] the quotient map of some sort or other, where [ilmath]i\in I[/ilmath] doesn't matter, as they're all the same under the quotient map.
- [ilmath]I:\eq[0,1]\subset\mathbb{R} [/ilmath]
- [math](\mathbb{S}^n,p)\cong\text{RS}\left(\mathbb{S}^{n-1},p\right):\eq\left(\frac{\mathbb{S}^{n-1}\times I}{(\{p\}\times I)\cup(\mathbb{S}^{n-1}\times\{0,1\})},\pi(p,i)\right)[/math] for [ilmath]\pi[/ilmath] the quotient map of some sort or other, where [ilmath]i\in I[/ilmath] doesn't matter, as they're all the same under the quotient map.
Notes
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Although not the best quotient we do have the situation on the right:
- By the characteristic property of the quotient topology:
- [ilmath]f\circ\pi[/ilmath] is continuous if and only if [ilmath]f[/ilmath] is continuous
That gives us an association between continuous maps of the form [ilmath]f\circ\pi[/ilmath] with some constraints. Blah blah blah, something like that.
Pointed topological spaces are involved.
