标题： 特殊酉群 $SU(2n)$ 作为框架流形 显示英文标题

标题： The special unitary groups $SU(2n)$ as framed manifolds

摘要： 设 $[SU(2n), \mathscr{L}]$表示带有其左不变框架 $SU(2n)$ $(n\ge 2)$ 的边群类 $\mathscr{L}$。 然后众所周知 $e_\mathbb{C}([SU(2n), \mathscr{L}])=0$，其中 $e_\mathbb{C}$表示复 Adams $e$-不变量。 在本文中，我们证明将$\mathscr{L}$替换为通过用特定映射对其扭转得到的框架，零值的$e_\mathbb{C}([SU(2n), \mathscr{L}])$可以转化为与循环群同构的$\mathrm{Im} \, e_\mathbb{C}$的生成元。此外，我们还证明同样的过程对于$SU(2n+1)$关于一个圆子群的商也给出了类似的结果，该商继承了来自$SU(2n+1)$的典型框架，方式与通常相同。 显示英文摘要

摘要： Let $[SU(2n), \mathscr{L}]$ denote the bordism class of $SU(2n)$ $(n\ge 2)$ equipped with its left invariant framing $\mathscr{L}$. Then it is well known that $e_\mathbb{C}([SU(2n), \mathscr{L}])=0$ where $e_\mathbb{C}$ denotes the complex Adams $e$-invariant. In this note we show that replacing $\mathscr{L}$ by the framing obtained by twisting it by a specific map the zero value of $e_\mathbb{C}([SU(2n), \mathscr{L}])$ can be transformed into a generator of $\mathrm{Im} \, e_\mathbb{C}$ which is isomorphic to a cyclic group. In addition we show that the same procedure affords an analogous result for a quotient of $SU(2n+1)$ by a circle subgroup which inherits a canonical framing from $SU(2n+1)$ in the usual way. .