I've just read the Bott Periodicity proof (complex case) in Hatcher book about K-theory.
At first it seems that using this theorem I could conclude that every complex vector bundles over $S^{2n+1}$ is trivial, but then, just handling the definition I couldn't prove that. In fact I don't think it is true.
Just handling the definitions we have that for every vector bundle $E$, $$E \sim 0$$ leads to $E \oplus n \approx m$ where $0,n,m$ are trivial bundles. So I was wondering if this relation has some visual consequences.
What is the interpretation of the Bott Periodicity with regards complex vector bundles over $S^{2n+1}$ and $S^{2n}$? Is only the fact that for every vector bundle $E$ $$E \oplus n \approx m$$ holds?
Just to be clear: by visual consequences I mean consequences that can be seen in $\text{Vect}_{\mathbb{C},n}(S^{2n})$ or $\text{Vect}_{\mathbb{C},n}(S^{2n+1})$
Yes, that's correct; we can only conclude that complex vector bundles over odd spheres are stably trivial, but not that they're trivial.
Using the version of Bott periodicity which states that the homotopy groups $\pi_i(U)$ of the stable unitary group are $2$-periodic, we see this as follows. Recall that for any topological group $G$ and any $n \ge 2$ we have $[S^n, BG] \cong \pi_{n-1}(G)$ (for $n = 1$ there is an issue with basepoints and the RHS is instead conjugacy classes in $\pi_0(G)$). In particular, $[S^n, BU]$, which describes isomorphism classes of stable complex vector bundles over $S^n$, is $\pi_{n-1}(U)$. By Bott periodicity this vanishes if $n$ is odd and is $\mathbb{Z}$ if $n$ is even.
(Using instead the version of Bott periodicity which is stated in terms of K-theory, the point is that the K-theory class of a vector bundle only sees the corresponding stable bundle.)
Bott (I think?) proved that $\pi_{2n}(U(n)) \cong [S^{2n-1}, BU(n)] \cong \mathbb{Z}_{n!}$, so the odd spheres $S^{2n-1}$ have nontrivial $n$-dimensional complex vector bundles on them for all $n \ge 2$.