A short note on $\pi_1\mathrm{Diff}_{\partial} (D^{4k})$ for $k\geq 3$

Let $\mathrm{Diff}_{\partial}(D^{n})$ be the topological group of diffeomorphisms of $D^{n}$ which agree with the identity near the boundary. In this short note, we compute the fundamental groups $\pi_1 \mathrm{Diff}_{\partial}(D^{4k})$ for $k\geq 3$. We also compute some Gromoll filtration groups $\Gamma^{4k+3}_3$ for $k\geq 3$.