Difference between revisions of "Nth homotopy group"
From Maths
(Created page with "{{DISPLAYTITLE:{{M|n^\text{th} }} homotopy group}} {{Stub page|grade=A*|msg=Not super urgent, but would be useful - demote to {{C|C}} once the content is less note-like}} __TO...") |
m |
||
| Line 1: | Line 1: | ||
{{DISPLAYTITLE:{{M|n^\text{th} }} homotopy group}} | {{DISPLAYTITLE:{{M|n^\text{th} }} homotopy group}} | ||
| + | : '''Note: ''' the [[fundamental group]] is {{M|\pi_1(X,p)}} | ||
{{Stub page|grade=A*|msg=Not super urgent, but would be useful - demote to {{C|C}} once the content is less note-like}} | {{Stub page|grade=A*|msg=Not super urgent, but would be useful - demote to {{C|C}} once the content is less note-like}} | ||
__TOC__ | __TOC__ | ||
| Line 29: | Line 30: | ||
** {{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. | ||
| + | |||
| + | |||
| + | [[Pointed topological space|Pointed topological spaces]] are involved. | ||
<div style="clear:both;"></div> | <div style="clear:both;"></div> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Topology|Algebraic Topology}} | {{Definition|Topology|Algebraic Topology}} | ||
Latest revision as of 22:17, 12 December 2016
- Note: the fundamental group is π1(X,p)
Stub grade: A*
This page is a stub
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:
Not super urgent, but would be useful - demote to C once the content is less note-like
Contents
[hide]Definition
Let (X,\mathcal{J}) be a topological space with x_0\in X being any fixed point. The n^\text{th} homotopy group, written \pi_n(X,x_0) is defined as follows:
- The underlying set of the group is: \pi_n(X,x_0):\eq[(\mathbb{S}^n,p),(X,x_0)]_*
- Where [(\mathbb{S}^n,p),(X,x_0)]_* denotes equivalence classes of continuous maps where f(p)\eq x_0 under the equivalence relation of homotopy relative to p, i.e.:
- [(\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)}
- Where [(\mathbb{S}^n,p),(X,x_0)]_* denotes equivalence classes of continuous maps where f(p)\eq x_0 under the equivalence relation of homotopy relative to p, 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:
- (\mathbb{S}^n,p)\cong\text{RS}\left(\mathbb{S}^{n-1}\right) where \text{RS} denotes the reduced suspension of a space we see:
- (\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) for \pi the quotient map of some sort or other, where i\in I doesn't matter, as they're all the same under the quotient map.
- I:\eq[0,1]\subset\mathbb{R}
- (\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) for \pi the quotient map of some sort or other, where i\in I doesn't matter, as they're all the same under the quotient map.
Notes
| |
| Diagram |
|---|
Although not the best quotient we do have the situation on the right:
- By the characteristic property of the quotient topology:
- f\circ\pi is continuous if and only if f is continuous
That gives us an association between continuous maps of the form f\circ\pi with some constraints. Blah blah blah, something like that.
Pointed topological spaces are involved.
