Volume distortion in homotopy groups
Volume distortion in homotopy groups
复制标题
同伦群中的体积畸变
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Fedor Manin
中科院分区：
文献类型：
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作者：
Fedor Manin
给定一个有限度量CW复形X和一个元素$${alpha in pi_n（X）}$$α∈πn（X），$${alpha}$$α的几何最优表示的性质是什么？我们研究了$${kalpha}$$kα作为k的函数的最优体积。渐近地，这个函数，其逆，由于传统的原因，我们称之为体积畸变，证明是关于X的有理同伦的不变量。我们提供了一些例子和技术来研究这个不变量，特别关注的空间与几个合理的同伦群。我们的主要定理刻画了X中所有的非挠同伦类都是无畸变的，即它们的畸变函数是线性的。
Given a finite metric CW complex X and an element $${alpha in pi_n(X)}$$α∈πn(X), what are the properties of a geometrically optimal representative of $${alpha}$$α? We study the optimal volume of $${kalpha}$$kα as a function of k. Asymptotically, this function, whose inverse, for reasons of tradition, we call the volume distortion, turns out to be an invariant with respect to the rational homotopy of X. We provide a number of examples and techniques for studying this invariant, with a special focus on spaces with few rational homotopy groups. Our main theorem characterizes those X in which all non-torsion homotopy classes are undistorted, that is, their distortion functions are linear.