Note on some Gromoll filtration groups and fundamental groups of $\mathrm{Diff}_{\partial}(D^n)$
Wei Wang
TL;DR
The article advances the understanding of the Gromoll filtration by explicitly computing several filtration groups $\Gamma^{n+1}_{i+1}$ in a range of dimensions and linking these to the homotopy groups of $\mathrm{Diff}_{\partial}(D^n)$. It merges two vantage points—Milnor–Novikov pairings (via plumbing and commutator constructions) and the Frank–Smith relationship—to place elements in the Gromoll filtration and determine their images under the $J$-homomorphism, aided by Morlet equivalences. The core results include precise identifications such as $\Gamma^{10}_{4}=\Gamma^{10}_{2}\cong \mathbb{Z}_2\oplus \mathbb{Z}_3$, $\Gamma^{13}_{7}=\Gamma^{13}_{2}\cong \mathbb{Z}_3$, and $\Gamma^{18}_{4}=\Gamma^{18}_{2}\cong \mathbb{Z}_8\oplus \mathbb{Z}_2$, together with several split surjectivity statements for $\pi_i\mathrm{Diff}_{\partial}(D^n)$ in low dimensions. The work also derives corollaries on $\pi_1\mathrm{Diff}_{\partial}(D^n)$ for $n$ in small ranges and proves a version of Frank–Smith's theorem relating boundary data, Toda brackets, and Thom-space attachments. Overall, the paper provides concrete filtration data and machinery linking exotic spheres, $J$-maps, and diffeomorphism groups, with implications for understanding high-dimensional diffeomorphism structures.
Abstract
In this note, we will compute some Gromoll filtration groups $Γ^{n+1}_{i+1}$ for certain $i$ when $8\leq n \leq 17$ and $n=4k+2\geq 18$. We will also use these results to obtain some information of $π_1\mathrm{Diff}_{\partial} (D^n)$ when $6\leq n \leq 15$ and $π_2 \mathrm{Diff}_{\partial} (D^{4k+3})$ when $4k+3\geq 15$.