Let $G$ be a compact Lie group. Let $H$ be a closed subgroup of $G$, and $K$ a closed subgroup of $H$. Let $Y$ be a $K-$space. Given a map in $Map_K(H, Y)$, I wonder how to extend it to a map in $Map_K(G, Y)$.
We may consider $Map_K(H, Y)$ as the space of section of the fibre bundle $$H\times_K Y\longrightarrow K\backslash H$$ with fibre $Y$, and $Map_K(G, Y)$ as the space of section of the fibre bundle $$G\times_H(H\times_K Y)\longrightarrow K\backslash G$$ with fibre $Y$ as well. Then How can we extend a section of the first bundle with smaller base to the section of the second one?
Thanks.
You can't always extend such a map. Indeed take $G = S^3$ and $H=K=S^1$. If you want $H \neq K$ take $H=O(2)$. There aren't any $S^1$-maps $G \to K$, because if $\varphi$ was such a map, then $\varphi^{-1}(e)$ would be (the image of) a section of the Hopf bundle $S^3 \to S^3/S^1 = S^2$. But this bundle is nontrivial, so has no sections. (In particular if you had a section you would be able to probe $S^3$ is homeomorphic to $S^2 \times S^1$!)