圆盘微分同胚的导数映射：一个例子
The derivative map for diffeomorphism of disks: An example
AI总结：
本文证明圆盘微分同胚的导数映射在 $k=11$ 时于同伦群上诱导非零映射，并据此给出 Burghelea–Lashof 猜想的反例。
摘要：
我们证明，通过对微分同胚取导数所定义的导数映射 $d \colon \mathrm{Diff}_\partial(D^k) \to Ω^kSO_k$，能够在同伦群上诱导出非平凡映射。具体而言，对于 $k=11$，我们证明以下同态非零：$d_* \colon π_5\mathrm{Diff}_\partial(D^{11}) \to π_5Ω^{11}SO_{11} \cong π_{16}SO_{11}$。作为推论，我们给出了 Burghelea 和 Lashof 一个猜想的反例，从而给出球面上一个非平凡向量丛 $E$ 的例子，而它作为拓扑 $\mathbb{R}^k$-丛是平凡的（$E$ 的秩为 $k=11$，底球面为 $S^{17}$）。证明依赖于 Burklund 和 Sencer 关于界定哪些同伦 17-球面以 8-连通流形为边界的最新结果、Antonelli、Burghelea 和 Kahn 对 Gromoll 滤过采用的 plumbbing 方法，以及某些同伦球面低余维嵌入的显式构造。
英文摘要：
We prove that the derivative map $d \colon \mathrm{Diff}_\partial(D^k) \to Ω^kSO_k$, defined by taking the derivative of a diffeomorphism, can induce a nontrivial map on homotopy groups. Specifically, for $k = 11$ we prove that the following homomorphism is non-zero: $$ d_* \colon π_5\mathrm{Diff}_\partial(D^{11}) \to π_{5}Ω^{11}SO_{11} \cong π_{16}SO_{11} $$ As a consequence we give a counter-example to a conjecture of Burghelea and Lashof and so give an example of a non-trivial vector bundle $E$ over a sphere which is trivial as a topological $\mathbb{R}^k$-bundle (the rank of $E$ is $k=11$ and the base sphere is $S^{17}$.) The proof relies on a recent result of Burklund and Senger which determines those homotopy 17-spheres bounding $8$-connected manifolds, the plumbing approach to the Gromoll filtration due to Antonelli, Burghelea and Kahn, and an explicit construction of low-codimension embeddings of certain homotopy spheres.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
