论文标题
循环空间中有效的周期
Efficient Cycles in Loop Space
论文作者
论文摘要
本文调查了riemannian歧管循环空间中一个循环的几何形状如何控制其拓扑。对于固定的$β\，在h^n（ωx; \ m athbb {r}）$中，一个人可以问一个$ | \ langle的β，z \ rangle | $可以适用于循环$ z $，以$ \ leq l $的循环和体积$ \ leq l^{n-1} $ for A in loq for Ane veledion for Ane velection for Anemaper的$ \ leq l $ ups $ \ leq l $ \ leq l $。我们表明，与此问题的上限提供了上限，这是Gromov对较高同型组的失真。我们还表明，我们可以表现出比目前因格罗莫夫失真相应问题而闻名的更好的下限。具体来说，我们表明存在一个$β$，检测$ [（\ Mathbb {cp}^2）^{\＃4} \ times s^2]^\ circ $的同型穿刺类别的同型$ | \langleβ，z \ \ \ \ \ \ \ rang | =ω（l^6/\ text {log} l）$。
This paper investigates how the geometry of a cycle in the loop space of a Riemannian manifold controls its topology. For fixed $β\in H^n(ΩX; \mathbb{R})$ one can ask how large $|\langle β, Z \rangle|$ can be for cycles $Z$ supported in loops of length $\leq L$ and of volume $\leq L^{n-1}$ for a suitably defined notion of volume of in loop space. We show that an upper bound to this question provides upper bounds Gromov's distortion of higher homotopy groups. We also show that we can exhibit better lower bounds than are currently known for the corresponding questions for Gromov's distortion. Specifically, we show there exists a $β$ detecting the homotopy class of the puncture in $[(\mathbb{CP}^2)^{\#4} \times S^2]^\circ$ and a family of cycles $Z_L$ with the geometric bounds above such that $|\langle β, Z \rangle| = Ω(L^6/\text{log}L)$.
