[ilmath]n^\text{th} [/ilmath] homotopy group

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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]

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]

Notes

[math]\xymatrix{ X \times I \ar@{.>}[dr]^{f\circ\pi} \ar[d]_\pi & \\ \frac{X\times I}{(\{p\}\times I)\cup(X\times \{0,1\})} \ar[r]_-f & Y }[/math]
Diagram

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.