$n^\text{th}$ homotopy group

Note: the fundamental group is $\pi_1(X,p)$

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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.:
    • $$[(\mathbb{S}^n,p),(X,x_0)]_*:\eq\frac{\{f\in C(\mathbb{S}^n,X)\ \vert\ f(p)\eq x_0\} }{\big({\small(\cdot)}\ \simeq\ {\small(\cdot)}\ (\text{Rel }\{p\}) \big)}$$

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.
    • $I:\eq[0,1]\subset\mathbb{R}$

Notes

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