MULTICOMPLEXES, BOUNDED COHOMOLOGY AND ADDITIVITY OF SIMPLICIAL VOLUME
MULTICOMPLEXES, BOUNDED COHOMOLOGY AND ADDITIVITY OF SIMPLICIAL VOLUME
复制标题
复形、有界上同调和单纯体积的可加性
DOI：
10.4134/bkms.2015.52.6.1855
复制
发表时间：
2001
影响因子：
0.5
通讯作者：
Thilo Kuessner
中科院分区：
文献类型：
--
作者：
Thilo Kuessner
我们讨论了带有边界的流形的单纯体积的一些可加性性质：我们给出了粘合合适的边界分量的可加性和粘合合适的边界子流形的超可加性的证明，并且我们讨论了 3-流形的加倍。本文致力于研究沿着合适的子流形剪切和粘贴时单纯体积的行为。 Gromov 在 [1] 中已经证明了这样的结果，对于闭流形（[1]，第 3 节），通过相当基本的论证，对于开流形（[1]，第 4 节），通过更高级的论证。本文的目的是使用更接近封闭情况所需参数的初等参数，为具有边界的紧流形情况编写完整的证明。单纯体积是紧流形的同伦不变量。它在 [1] 中定义，使用奇异链上的 l-范数，如下所示。定义：a) 令 (X,Y ) 为一对空间，h ∈ H* (X,Y ;R) 为相对同调类。其格罗莫夫范数 ‖ h ‖ 定义为
We discuss some additivity properties of the simplicial volume for manifolds with boundary: we give proofs of additivity for glueing amenable boundary components and of superadditivity for glueing amenable submanifolds of the boundary, and we discuss doubling of 3-manifolds. This paper is devoted to the behaviour of simplicial volume under cutting and pasting along amenable submanifolds. Such results have been proved by Gromov in [1], for closed manifolds ([1], Section 3) by fairly elementary arguments, and for open manifolds ([1], Section 4) by much more advanced arguments. The aim of this article is to write complete proofs for the case of compact manifolds with boundary, using elementary arguments which are closer to the arguments that were needed for the closed case. The simplicial volume is a homotopy invariant of compact manifolds. It was defined in [1], using the l-norm on singular chains, as follows. Definition: a) Let (X,Y ) be a pair of spaces, and h ∈ H∗ (X,Y ;R) a relative homology class. Its Gromov norm ‖ h ‖ is defined by