Gromov的多重复合体理论及其在有界上同调和简单体积上的应用

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS ACS Applied Bio Materials Pub Date : 2018-08-22 DOI:10.1090/memo/1402
R. Frigerio, M. Moraschini
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引用次数: 24

摘要

简单体积是由Gromov在其开创性论文《体积与有界上同调》中引入的流形的同伦不变量。为了研究简单体积的主要性质,Gromov本人提出了有界上同的对偶理论,并发展成为一个非常活跃和独立的研究领域。Gromov的拓扑空间有界上同调理论是基于多重复形的使用,多重复形是一种简单结构,它推广了简单复形,但不允许在简单集合中出现所有的简并。本文的第一个目的是为多元配合物理论奠定基础。在确定了主要定义之后,构造了与拓扑空间X X相关的奇异复形K (X) \mathcal {K}(X),并证明了K (X) \mathcal {K}(X)的几何实现对于每一个CW复形X X都等价于X X。继Gromov之后,我们引入完备性的概念,粗略地说,它将简单集合的Kan条件转化为多重复形的背景。然后,我们发展了完全复复的同伦理论,并证明了K (X) \数学{K}(X)对于每一个CW复X X是完全的。在本工作的第二部分,我们将复复理论应用于拓扑空间的有界上同伦的研究。我们的构造和论证最终证明了Gromov映射定理(它特别暗示了空间的有界上同调只依赖于它的基本群)和Gromov消失定理,它保证了闭流形的简单体积的消失,允许小复数的可服从覆盖。本文的第三部分也是最后一部分研究了非紧空间上的局部有限链,从而研究了开流形的简单体积。我们扩展了Gromov的一些思想,给出了开流形简单体积的消失判据和有限判据的详细证明。作为这些结果的副产品,我们证明了在可服从覆盖下闭流形的1 \ well ^1 -不可见性的一个判据。作为应用,我们第一次详细地证明了三开流形积简体积的消失性。
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Gromov’s Theory of Multicomplexes with Applications to Bounded Cohomology and Simplicial Volume
The simplicial volume is a homotopy invariant of manifolds introduced by Gromov in his pioneering paper Volume and bounded cohomology. In order to study the main properties of simplicial volume, Gromov himself initiated the dual theory of bounded cohomology, which then developed into a very active and independent research field. Gromov’s theory of bounded cohomology of topological spaces was based on the use of multicomplexes, which are simplicial structures that generalize simplicial complexes without allowing all the degeneracies appearing in simplicial sets. The first aim of this paper is to lay the foundation of the theory of multicomplexes. After setting the main definitions, we construct the singular multicomplex K ( X ) \mathcal {K}(X) associated to a topological space X X , and we prove that the geometric realization of K ( X ) \mathcal {K}(X) is homotopy equivalent to X X for every CW complex X X . Following Gromov, we introduce the notion of completeness, which, roughly speaking, translates into the context of multicomplexes the Kan condition for simplicial sets. We then develop the homotopy theory of complete multicomplexes, and we show that K ( X ) \mathcal {K}(X) is complete for every CW complex X X . In the second part of this work we apply the theory of multicomplexes to the study of the bounded cohomology of topological spaces. Our constructions and arguments culminate in the complete proofs of Gromov’s Mapping Theorem (which implies in particular that the bounded cohomology of a space only depends on its fundamental group) and of Gromov’s Vanishing Theorem, which ensures the vanishing of the simplicial volume of closed manifolds admitting an amenable cover of small multiplicity. The third and last part of the paper is devoted to the study of locally finite chains on non-compact spaces, hence to the simplicial volume of open manifolds. We expand some ideas of Gromov to provide detailed proofs of a criterion for the vanishing and a criterion for the finiteness of the simplicial volume of open manifolds. As a by-product of these results, we prove a criterion for the ℓ 1 \ell ^1 -invisibility of closed manifolds in terms of amenable covers. As an application, we give the first detailed proof of the vanishing of the simplicial volume of the product of three open manifolds.
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来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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