[论文解读] The Kuenneth formula for graphs

Oliver Knill|arXiv (Cornell University)|May 28, 2015

Homotopy and Cohomology in Algebraic Topology参考文献 15被引用 5

一句话总结

本文引入了一种有限简单图的新型笛卡尔积，该积在图上同调中满足Künneth公式，确保乘积的Poincaré多项式可分解为各因子多项式的乘积。该构造利用离散de Rham上同调与链同伦，建立了de Rham上同调与单纯上同调之间的同构，证明了乘积图的上同调可分解为各因子上同调的张量积，其维数与色数性质与连续拓扑中的情形一致。

ABSTRACT

We construct a Cartesian product G x H for finite simple graphs. It satisfies the Kuenneth formula: H^k(G x H) is a direct sum of tensor products H^i(G) x H^j(G) with i+j=k and so p(G x H,x) = p(G,x) p(H,y) for the Poincare polynomial p(G,x) and X(G x H) = X(G) X(H) for the Euler characteristic X(G)=p(G,-1). G1=G x K1 has as vertices the simplices of G and a natural digraph structure. We show that dim(G1) is larger or equal than dim(G) and G1 is homotopic to G. The Kuenneth identity is proven using Hodge describing the harmonic forms by the product f g of harmonic forms of G and H and uses a discrete de Rham theorem given by a combinatorial chain homotopy between simplicial and de Rham cohomology. We show dim(G x H) = dim(G1) + dim(H1) implying that dim(G x H) is larger or equal than dim(G) + dim(H) as for Hausdorff dimension in the continuum. The chromatic number c(G1) is smaller or equal than c(G) and c(G x H) is bounded above by c(G)+c(H)-1. The automorphism group of G x H contains Aut(G) x Aut(H). If G~H and U~V then (G x U) ~ (H x V) if ~ means homotopic: homotopy classes can be multiplided. If G is k-dimensional geometric meaning that all unit spheres S(x) in G are (k-1)-discrete homotopy spheres, then G1 is k-dimensional geometric. If G is k-dimensional geometric and H is l-dimensional geometric, then G x H is geometric of dimension (l+k). The product extends to a ring of chains which unlike the category of graphs is closed under boundary operation taking quotients G/A with A subset Aut(G). As we can glue graphs or chains, joins or fibre bundles can be defined with the same features as in the continuum, allowing to build isomorphism classes of bundles.

研究动机与目标

• 定义一种图积，使其在离散上同调中满足Künneth公式，类比于连续情形。
• 利用偏导数与调和形式，在乘积图上建立离散de Rham上同调理论。
• 证明乘积图的上同调同构于各因子上同调的张量积。
• 将该积扩展为图上的环结构，对边界运算与商运算封闭，从而可构造图丛。

提出的方法

• 通过$G$和$H$中完全子图的对作为顶点，定义有限简单图$G \times H$的笛卡尔积。
• 利用$G$和$H$的偏导数构造$G \times H$上的离散外微分，形成de Rham复形。
• 在de Rham复形与Whitney单纯复形之间建立链同伦，证明两种上同调理论的等价性。
• 利用$G$和$H$上的调和形式$f$与$g$，在$G \times H$上生成基形式$f(x) \otimes 1$、$1 \otimes g(y)$与$f(x)g(y)$。
• 通过显式构造上同调代表元，证明Künneth公式$H^k(G \times H) \cong \bigoplus_{i+j=k} H^i(G) \otimes H^j(H)$。
• 证明$G \times H$的维数在每一点上为$G$与$H$维数之和，且$G_1 = G \times K_1$同伦于$G$。

实验结果

研究问题

• RQ1 是否存在一种图积，使其在图上同调中满足Künneth公式，类比于连续情形？
• RQ2 能否通过链同伦使乘积图上的离散de Rham上同调与单纯上同调等价？
• RQ3 乘积图的维数与各因子维数有何关系？是否满足类似Hausdorff的不等式？
• RQ4 图$G \times H$的色数是多少？是否满足$c(G \times H) \leq c(G) + c(H) - 1$？
• RQ5 该积构造能否扩展为对边界运算与自同构群作用的商运算封闭的图环？

主要发现

• 乘积图的Poincaré多项式满足$p_{G \times H}(x) = p_G(x) p_H(x)$，且$\chi(G \times H) = \chi(G) \chi(H)$。
• 乘积图的上同调同构于$\bigoplus_{i+j=k} H^i(G) \otimes H^j(H)$，满足Künneth公式。
• 乘积图的维数在每一点上为$G$与$H$维数之和：$\dim(G \times H)(x,y) = \dim(G_1)(x) + \dim(H_1)(y)$。
• 图$G \times H$的色数至多为$c(G) + c(H) - 1$，且当$G$与$H$均为3-可染色并包含$K_5$子图时，$c(G \times H) = 5$。
• 图$G \times H$的自同构群包含$\operatorname{Aut}(G) \times \operatorname{Aut}(H)$，且该积保持同伦类。
• 该积可扩展为一个链环，对边界运算$\delta$与商$G/A$（其中$A \subset \operatorname{Aut}(G)$）封闭，从而可构造图丛。