I am following up to this question. The setup is a fibre bundle $F \hookrightarrow E \to B$ such that $F \hookrightarrow E$ is nullhomotopic.
One starts with a map $g:S^{n-1} \to F$. This lifts to a map $\tilde{g}:D^n \to E$. (One can construct this explicitly; I did, but don't think it's necessary.) This $\tilde{g}$ represents a homotopy class in $\pi_n(E,F) \cong \pi_n(B)$.
I want to show that $[g] \mapsto [\tilde{g}]$ is well-defined, in fact a homomorphism. Right now I am still stuck on well-definedness.
Well-definedness
Let $g,h$ be homotopic maps $S^{n-1} \to F$. Then $\tilde{g},\tilde{h}$ are maps $(D^n,S^{n-1}) \to (E,F)$. Denote by $G:S^{n-1} \times I \to F$ the homotopy between $g,h$. We want a homotopy $\tilde{G}:D^n \times I \to E$ between $\tilde{g},\tilde{h}$. I thought to use the HLP; but for this, we need a homotopy $D^n \times I \to B$, whose construction is not obvious to me. How is this homotopy obtained? (Is the HLP even the right tool for this?)
Update: I think I was correct in that the HLP is too much for this problem. Let $H:F \times I \to E$ be the nullhomotopy from $F$ to $*$. Then define a homotopy $D^n \times I \to E$ by $(x,t) \mapsto H(G(x/|x|,t),|x|)$ and $(0,t) \mapsto *$. This is a perfectly good homotopy between lifts of $g,h$ to maps $(D^n,S^{n-1}) \to (E,F)$. This suffices.
Homomorphism
I am not sure how to begin this part, and am working on it now. Any help would be appreciated.
