论文标题
束束缚组的界限
Boundedness of bundle diffeomorphism groups over a circle
论文作者
论文摘要
在本文中，我们研究了一个圆圈束差异群的界限。对于纤维束$π：m \至s^1 $，带有纤维$ n $和结构组$γ$和$ r \ in {\ bbb z} _ {\ geq 0} \ cup \ cup \ {\ cup \ {\ infty \} $，我们区分了一个integer $ k = k = k = k = k = k = k = k = k = k（q，r）一个函数$ \wideHatν：{\ rm diff}_π（m）_0 \ to {\ bbb r} _k $。当$ k \ geq 1 $时，可以表明捆绑捆绑数差异$ {\ rm diff}_π（m）_0 $均匀完美，$clb_π\，{\ rm diff}^r_π（m）_0 \ leq k+3 $，如果非常适合带有纤维$ n $和结构组$γ$的微不足道的光纤束$ρ：e \ to {\ bbb r} $。另一方面，当$ k = 0 $时，表明$ \widehatν$是无限的准晶体，因此$ {\ rm diff}_π（m）_0 $是无界的，并且不是均匀的完美。我们还描述了映射圆圈$π的附件$ ϕ $在s^1 $的映射torus $π中的整数$ k $，并给出了一些（UN）有限组的明确示例。
In this paper we study boundedness of bundle diffeomorphism groups over a circle. For a fiber bundle $π: M \to S^1$ with fiber $N$ and structure group $Γ$ and $r \in {\Bbb Z}_{\geq 0} \cup \{ \infty \}$ we distinguish an integer $k = k(π, r) \in {\Bbb Z}_{\geq 0}$ and construct a function $\widehatν : {\rm Diff}_π(M)_0 \to {\Bbb R}_k$. When $k \geq 1$, it is shown that the bundle diffeomorphism group ${\rm Diff}_π(M)_0$ is uniformly perfect and $clb_π\,{\rm Diff}^r_π(M)_0 \leq k+3$, if ${\rm Diff}_{ρ, c}(E)_0$ is perfect for the trivial fiber bundle $ρ: E \to {\Bbb R}$ with fiber $N$ and structure group $Γ$. On the other hand, when $k = 0$, it is shown that $\widehatν$ is a unbounded quasimorphism, so that ${\rm Diff}_π(M)_0$ is unbounded and not uniformly perfect. We also describe the integer $k$ in term of the attaching map $ϕ$ for a mapping torus $π: M_ϕ\to S^1$ and give some explicit examples of (un)bounded groups.
