图的层及其同调不变量
Sheaves on Graphs and Their Homological Invariants
摘要：
我们引入了一个概念的一层向量空间的图，并发展的基础同调论这样的层。一个层不变量，它的“最大过剩”，有许多显着的性质。它有一个简单的定义，没有提到同调论，类似于图的扩展。然而，它是贝蒂数的“极限”，因此有一个短/长正合序列理论，类似于阿蒂亚的$L^2$贝蒂数。此外，最大超额是通过一个超模函数定义的，它给出了比典型的贝蒂数更强的最大超额性质。最大过剩给出了一个简单的解释一个重要的图不变量，这将被用来研究汉娜诺依曼猜想在未来的文件。我们的层理论可以被看作是代数图论的一个巨大的推广：每个层都有与之相关的不变量-如贝蒂数和拉普拉斯矩阵-这些不变量推广了经典图论中的不变量。
英文摘要：
We introduce a notion of a sheaf of vector spaces on a graph, and develop the foundations of homology theories for such sheaves. One sheaf invariant, its "maximum excess," has a number of remarkable properties. It has a simple definition, with no reference to homology theory, that resembles graph expansion. Yet it is a "limit" of Betti numbers, and hence has a short/long exact sequence theory and resembles the $L^2$ Betti numbers of Atiyah. Also, the maximum excess is defined via a supermodular function, which gives the maximum excess much stronger properties than one has of a typical Betti number. The maximum excess gives a simple interpretation of an important graph invariant, which will be used to study the Hanna Neumann Conjecture in a future paper. Our sheaf theory can be viewed as a vast generalization of algebraic graph theory: each sheaf has invariants associated to it---such as Betti numbers and Laplacian matrices---that generalize those in classical graph theory.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
