The connection Laplacian L and the Dirac matrix D are both n x n matrices defined from a given finite simplicial complex G with n sets. In both cases, there is interlacing of the eigenvalues for subcomplexes. This gives general upper bounds of the eigenvalues both for L and D in terms of inclusion or intersection degrees. We conjecture that L always dominates both D and the inverse of L in a weak Loewner sense. In a second part we look at dynamical systems (G,T), where T is a simplicial map on G. Both L and D generalize to dynamical versions of L and D. The modified L is still unimodular with an explicit Green function inverse and modified Dirac part still comes from an exterior derivative d. We also review the Lefschetz fixed point theorem for a simplicial map T on a simplicial complex G which implies the Brouwer fixed point theorem: any simplicial map on a contractible finite abstract simplicial complex G has a fixed simplex.

翻译：连接拉普拉斯算子 L 与狄拉克矩阵 D 均为 n×n 矩阵，其定义基于一个具有 n 个集合的给定有限单纯复形 G。在这两种情形下，子复形的特征值均存在交错现象。这为 L 和 D 的特征值提供了基于包含度或交度的通用上界。我们猜想，在弱勒夫纳意义下，L 始终同时支配 D 与 L 的逆矩阵。在第二部分中，我们考察动力系统 (G,T)，其中 T 是 G 上的单纯映射。L 与 D 均可推广至其动力系统版本。修正后的 L 仍为幺模矩阵，并具有显式的格林函数逆；修正后的狄拉克部分仍源于外导数 d。此外，我们回顾了单纯复形 G 上单纯映射 T 的莱夫谢茨不动点定理，该定理蕴含布劳威尔不动点定理：任何在可缩有限抽象单纯复形 G 上的单纯映射均存在一个不动单纯形。