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

TL;DR

This paper computes the fundamental groups of the diffeomorphism groups of high-dimensional disks, specifically for dimensions divisible by four and greater than or equal to 12, advancing understanding of their topological structure.

Contribution

It provides the first explicit calculation of the fundamental groups of these diffeomorphism groups for dimensions $4k$ with $k extgreater=3$, filling a gap in high-dimensional topology.

Findings

Computed $\pi_1 ext{Diff}_\partial(D^{4k})$ for $k extgreater=3$

Established new results on the topology of diffeomorphism groups in high dimensions

Enhanced understanding of the structure of diffeomorphism groups of disks

Abstract

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 $k\geq 3$.