论文标题
BING涉及的几何形状
The Geometry of the Bing Involution
论文作者
论文摘要
1952年,Bing出版了3秒$ s^3 $的狂野(不是拓扑结合到平滑的)$ i $。但是在分析上,它到底有多狂野?我们证明,从拓扑结合到$ i $的任何互动$ i^h $都必须具有连续性的几乎指数模量。具体而言,考虑到任何$α> 0 $,存在$δ$的序列,$δ> 0 $,点$ x,y \ in s^3 $,带有dist $(x,y)<δ$,但dist $(i^h(i^h(x),i^h(y),i^h(y),i^h(y)) e^{\ left(\ frac {ε^{ - 1}} {\ log^{(1+α)}(ε^{ - 1})} \ right)} $,dist是$ s^3 $的常规riemannian距离。尤其是,$ i^h $伸展距离要比Lipschitz函数($δ^{ - 1} =cε^{ - 1} $)或hölder函数($δ^{ - 1} = c^\ prime(c^\ prime(ε^{ - 1}) Bing的原始结构和已知替代方案(请参见文本),$ i $具有连续性的模量$δ^{ - 1}> c \ sqrt {2}^{ε^{ - 1}} $,因此定理是相当紧密的 - 我们必须表明,至少必须完全实现polyential expectiential offectepential expectient to polylog,而事实是事实。实际上,可以选择$δ^{ - 1} $的功能,可以选择比此处所述的指数略接近指数(请参见Theorem 1)。使用相同的技术,我们分析了一大批``'''bing的参与,并表明,作为一项学者,它们可以将任何功能$ f:\ mathbb {r}^+ \ rightarrow \ rightarrow \ mathbb {r}^+ $,无论其增长如何快速,我们都可以找到$ j $ j $ j $ j $ j $ j $ j^$ j^$ j^$ j^$ j^$ j^$ j^$ j.连续性的模量$δ^{ - 1}(ε^{ - 1})$生长速度快于$ f $(近无限)。有关于固有的可不同性(文本中的参考文献)的文献,但据作者知道,连续性固有模量的主题是新的。
In 1952 Bing published a wild (not topologically conjugate to smooth) involution $I$ of the 3-sphere $S^3$. But exactly how wild is it, analytically? We prove that any involution $I^h$, topologically conjugate to $I$, must have a nearly exponential modulus of continuity. Specifically, given any $α>0$, there exists a sequence of $δ$'s converging to zero, $δ> 0$, and points $x,y \in S^3$ with dist$(x,y) < δ$, yet dist$(I^h(x), I^h(y)) > ε$, where $δ^{-1} = e^{\left(\frac{ε^{-1}}{\log^{(1+α)}(ε^{-1})}\right)}$, and dist is the usual Riemannian distance on $S^3$. In particular, $I^h$ stretches distance much more than a Lipschitz function ($δ^{-1} = cε^{-1}$) or a Hölder function ($δ^{-1} = c^\prime(ε^{-1})^{p}$, $1 < p < \infty$). Bing's original construction and known alternatives (see text) for $I$ have a modulus of continuity $δ^{-1} > c \sqrt{2}^{ε^{-1}}$, so the theorem is reasonably tight -- we prove the modulus must be at least exponential up to a polylog, whereas the truth may be fully exponential. Actually, the functional for $δ^{-1}$ coming out of the proof can be chosen slightly closer to exponential than stated here (see Theorem 1). Using the same technique we analyze a large class of ``ramified'' Bing involutions and show, as a scholium, that given any function $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$, no matter how rapid its growth, we can find a corresponding involution $J$ of the 3-sphere such that any topological conjugate $J^h$ of $J$ must have a modulus of continuity $δ^{-1}(ε^{-1})$ growing faster than $f$ (near infinity). There is a literature on inherent differentiability (references in text) but as far as the authors know the subject of inherent modulus of continuity is new.