标题： 零同伦体积在幂零空间中 显示英文标题

标题： Volumes of Nullhomotopies in Nilpotent Spaces

摘要： 曼因的影子原理已被证明是解决格罗莫夫在20世纪末提出的定量拓扑问题的有用工具。该原理非正式地提供了一种方法，将微分分次代数之间的有界代数映射转换为它们几何实现之间的附近真实映射。我们将这一原理扩展到主$K(G,n)$纤丛的有限塔，并特别将此构造应用于幂零空间。作为扩展原理的具体应用，我们提供了到幂零空间中的Lipschitz映射的同伦体积渐近行为的上界。当$c = 1$时，我们进一步改进这些界限，几乎达到单连通情况下的界限。在目标空间为共形的情况下，我们也类似地改进这些界限，并证明在这种情况下界限几乎是精确的。 显示英文摘要

摘要： The Shadowing Principle of Manin has proved a valuable tool for addressing questions of quantitative topology raised by Gromov in the late 1900s. The principle informally provides a way for bounded algebraic maps between differential graded algebras to be translated into nearby genuine maps between their geometric realizations. We extend this principle to finite towers of principal $K(G,n)$ fibrations, and in particular apply this construction to nilpotent spaces. As a specific application of the extended principle, we provide upper bounds on the asymptotic behavior of volumes of nullhomotopies of Lipschitz maps into nilpotent spaces. We further refine these bounds in the case when $c = 1$ to nearly meet those of the simply connected setting. We similarly refine these bounds in the event the target space is coformal, and demonstrate that the bounds in this setting are nearly sharp.