Note on some Gromoll filtration groups and fundamental groups of ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\mathrm{\partial }}(D^n)$

In this note, we will compute some Gromoll filtration groups ${\mathrm{\Gamma }}_{i+1}^{n+1}$ for certain $i$ when $8\le n\le 17$ and $n=4k+2\ge 18$. We will also use these results to obtain some information of ${\pi }_1{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\mathrm{\partial }}(D^n)$ when $6\le n\le 15$ and ${\pi }_2{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}_{\mathrm{\partial }}(D^{4k+3})$ when $4k+3\ge 15$.