I am trying to understand how to use Hopf invariant, to calculate $\pi_{4n-1}(S^{2n})$.
I've started with defining a new space $X$ adjoining $D^{4n}$ to $S^{2n}$ via a map $\phi\in\pi_{4n-1}(S^{2n}) $. I can see that this new space has $C_n(X)$ zero when $n$ is not 0, 2n, 4n.
Now I need to calculate $H^{4n}$, $H^{2n}$. For this do I need to use some kind of exact sequence of cohomology groups?
Then I need to show that generators of these groups are one is multiple of the other using cup product.
Lastly, I need to show this multiple element, called Hopf invariant, only depends on homotopy class of $\phi$.
This theorems are stated in algebraic topology books however I am struggling to understand mid-steps. Any help, hint or idea will be appreciated.