# Computing the homotopy groups $\pi_n(G,G_q)$

Let $$G_q$$ be the space of degree one maps $$S^{q-1}\rightarrow S^{q-1}$$ and $$G=lim G_q$$ under suspensions $$G_q\subset G_{q+1}$$. What are the known computations of $$\pi_n(G,G_q)$$?
I found that $$\pi_3(G,G_3)=\mathbb{Z}_2$$ and $$\pi_4(G,G_3)=\mathbb{Z}$$ in Haefliger's https://doi.org/10.2307/1970475 section 5.16.

By Theorem A of Milgram's On the Haefliger knot groups, there is a $$(2k-3)$$-connected map
$$\mathrm{SO}/\mathrm{SO}(k) \longrightarrow \mathrm{G}/\mathrm{G}_k,$$