I am reading about some of the historical motivations leading up to the discovery of the Hopf Fibration in 1931, but I am having some trouble with some intuition behind why the map was such a shock to the math community.
Apparently, the argument posed by mathematicians as to why such a map could not exist was by analogy using homology groups.
The Hopf fibration was the first example of a map $\varphi:S^{n+1}\rightarrow S^n$ that was not null-homotopic. My thinking is as follows:
The homology classes of $S^n$ are given by: $$ H_k(S^n) = \begin{cases} \mathbb Z & k=0,n \\ \{0\} & \text{otherwise} \end{cases} $$ So clearly, $H_{n+1}(S^n) = \{0\}$. Is this the reason mathematicians thought that $\pi_{n+1}(S^n)=\{0\}$, and in particular for the Hopf fibration that every map $f:S^3\rightarrow S^2$ would be null-homotopic? Or is my understanding off?