Could anyone help me to prove $$\pi_n(S^n) \quad \text{is homomorphic to}\quad \mathbb{Z}.$$
There is so many solutions in books with the help of homp theory.
But I want to prove it directly without help of any other theory. I think maybe I could prove it by induction. Is it ok ?
I have found a proof form the book Homology Theory. But I’m not sure about it. It says :
Example. Let $<\alpha>$ be a class in $\pi_1(S^1,x_0)$, where $x_0$ is chosen to be the point $(1,0)$ in $S^l$. Using the covering space $exp: \mathbb{R} \rightarrow S^1$, we lift $\alpha$ to a path in $\mathbb{R}$ with initial point $0$. The terminal point of this lift is an integer which we denote $d(\alpha)$. Since any loop homotopic to $\alpha$ must lift to a path with the same terminal point, $d(\alpha)$ depends only on the class of $\alpha$ in $\pi_1(S^1,x_0)$.Consequently, $d$ defines a function from $\pi_1(S^1,x_0)$ to $\mathbb{Z}$. Note that if the initial point of the lift of $\alpha$ is taken to be the integer $k$, then the terminal point of the lift will be $k + d(\alpha)$. Consequently, if $\alpha$ and $\beta$ are loops at $x_0$, then $d(\alpha \cdot \beta) = d(\alpha) + d(\beta)$. In other words, we have produced a homomorphism $$d:\pi_1(S^1,x_0) \longrightarrow \mathbb{Z}, $$ called the degree of the loop.
For any integer $m$ in $\mathbb{Z}$ there is a path $\widetilde{\gamma}$ from $0$ to $m$ in $\mathbb{R}$. Projecting this path down to the base, $\gamma = exp\widetilde{\gamma}$ is a loop at $x_0$ for which $d(\gamma) = m$; hence $d$ is an epimorphism. On the other hand, let $\alpha$ and $\beta$ be loops in $S^1$ with $d(\alpha) = k = d(\beta)$. So the lifts $\widetilde{\alpha}$ and $\widetilde{\beta}$ are paths in $\mathbb{R}$ with initial point $0$ and terminal point $k$. Define a function $$H: [0,1] \times [0,1] \longrightarrow \mathbb{R} $$ by $H(t,s) = (1-s)\widetilde{\alpha}(t) + s\widetilde{\beta}(t), 0 \geq s \geq 1$. Since $\mathbb{R}$ is convex, this is well defined and continuous. This homotopy from $\widetilde{\alpha}(s = 0)$ to $\widetilde{\beta}(s = 1)$ fixes the endpoints at $0$ and $k$ throughout the deformation. Then $exp H$ is a based homotopy from $\alpha$ to $\beta$. Therefore $<\alpha>=<\beta>$, and $d$ is a monomorphism. This completes the proof of the following proposition.
4.6 Proposition. The degree of a loop defines an isomorphism $$d:\pi_1(S^1,x_0) \longrightarrow \mathbb{Z}.$$
Specifically I look for the special case of $n=1, 2, 3, 4,... $
There is a proof in the book Nonabelian Algebraic Topology p.269. It derives from a Suspension Theorem which says that if the space $A$ is $(n-2)$-connected for $n\geqslant 3$ then its suspension $SA$ is $(n-1)$-connected and $\pi_n(SA) \cong \pi_{n-1}(A)$. However this itself is deduced from a higher order Seifert-van Kampen Theorem, whose main applications are given in Chapter 8 of the book, and whose proof is not easy; so I do not really expect that this approach meets the (rather hard) criteria! The proof though does not use singular homology.