Let $X$ be an $H$ space and $\tilde{H}$ it's universal cover. It is well known that the action of $\pi_1(H)$ on $\pi_n(H)$ is trivial. I am seeking an alternative proof of this fact using the universal cover.
Amongst the possible definitions of the action of $\pi_1(H)$ one is to use the the isomorphisms $\pi_1(H)\cong Deck(\tilde{H}\to H)$ and $\pi_n(H)\cong \pi_n(\tilde{H})$. So, dreaming of some equivariance, in theory I should be able to see that the action of $\pi_1(H)$ on $\pi_n(H)$ is trivial by studying the action of $Deck(\tilde{H}\to H)$ on $\pi_n(\tilde{H})$.
I haven't been able to show this, nor have I even been able to show the desired equivariance. Any help is appreciated.
Thankss