论文标题
磁盘差异性的衍生图图：一个例子
The derivative map for diffeomorphism of disks: An example
论文作者
论文摘要
我们证明，派生映射$ d \ colon \ mathrm {diff} _ \ partial（d^k）\ toω^kSO_K $，通过采用差异性的衍生物来定义，可以在同型组上诱导非繁琐的映射。具体而言，对于$ k = 11 $，我们证明以下同构为非零：$$ d_* \cOLONπ_5\ mathrm {diff} _ \ partial（d^{11}）反示例与Burghelea和Lashof的猜想相反，因此给出一个非平凡的矢量束$ e $的示例，而不是一个小的球体，这是一个拓扑$ \ Mathbb {r}^k $ -bundle（$ e $的等级$ k = 11 $，而base s $ k = 11 $，base spheys is base spheys is base s is $ s^17} $ s^{17} $）。 证明取决于伯克伦德和森格的最新结果，该结果决定了这些同质拷贝的17范围，限制了$ 8 $连接的歧管，由于安东利（Antonelli），伯格利亚（Burghelea）和卡恩（Kahn）而引起的gromoll过滤的管道方法，以及某些同型spheres spheres spheres semotsimense嵌入的明确结构。
We prove that the derivative map $d \colon \mathrm{Diff}_\partial(D^k) \to Ω^kSO_k$, defined by taking the derivative of a diffeomorphism, can induce a nontrivial map on homotopy groups. Specifically, for $k = 11$ we prove that the following homomorphism is non-zero: $$ 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.
