We prove a Lefschetz formula for general simple graphs which equates the Lefschetz number L(T) of an endomorphism T with the sum of the degrees i(x) of simplices in G which are fixed by T. The degree i(x) of x with respect to T is defined as a graded sign of the permutation T induces on the simplex x multiplied by -1 if the dimension of x is odd. The Lefschetz number is defined as in the continuum as the super trace of T induced on cohomology. In the special case where T is the identity, the formula becomes the Euler-Poincare formula equating combinatorial and cohomological Euler characteristic. The theorem assures in general that if L(T) is nonzero, then T has a fixed clique. A special case is a discrete Brouwer fixed point theorem for graphs: if T is a graph endomorphism of a connected graph G, which is star-shaped in the sense that only the zeroth cohomology group is nontrivial, like for connected trees or triangularizations of star shaped Euclidean domains, then there is clique x which is fixed by T. Unlike in the continuum, the fixed point theorem proven here looks for fixed cliques, complete subgraphs which play now the role of "points" in the graph. Fixed points can so be vertices, edges, fixed triangles etc. If A denotes the automorphism group of a graph, we also look at the average Lefschetz number L(G) which is the average of L(T) over A. We prove that this is the Euler characteristic of the graph G/A and especially an integer. We also show that as a consequence of the Lefschetz formula, the zeta function zeta(T,z) is a product of two dynamical zeta functions and therefore has an analytic continuation as a rational function which is explicitly given by a product formula involving only the dimension and the signature of prime orbits of simplices in G.
Handel's fixed point theorem: A Morse theoretical point of view
Michael Handel在[Ha]中证明了开单位圆盘的保向同胚的一个不动点定理，它被证明是研究曲面同胚动力学的一个有效工具。本文符合作者[LeC 2]和Juliana Xavier [X1]，[X2]的一系列文章，其中给出了与经典Brouwer理论相关的证明，而不是原始文章中使用的同伦Brouwer理论。与[LeC2]、[X1]和[X2]一样，我们将使用“自由砖分解”，但将提出一个更概念化的莫尔斯理论论证。它基于一个新的初步引理，为我们的问题提供了一个很好的“无穷大条件”。
对于每一个布劳威尔（即平面，不动点自由，方向保持）同胚h存在一个覆盖的平面平移域，不变的简单连通开子集上的h是共轭仿射平移。我们在平面上引入一个距离d_h，它计算连接一对点的平移域的最小数目。这使我们能够描述一个组合共轭不变量，并证明存在一个有限族的广义Reeb分量分离任何两个点x，y，使d_h（x，y）>1。Brouwer's 'obtient en translation domaines（ouverts simplement connexes，invariants，en restriction auxquels la dynamique est conjuguee a une translation）的同胚论。本文介绍了一种计算两点不重合的平移域最小值的平面上的距离d_h。我们可以定义一个不变的共轭组合，它定义了翻译领域中的三大类。关于动态结构的存在的一种平等性，即在非平凡的计划中普遍存在着Reeb的复合物。
迈克尔·亨德尔在[7]中证明了一个不动点的存在性，为一个方向保持同胚的开放单位磁盘，可以扩展到封闭的磁盘，只要它有点的轨道形成一个有向循环的联系在无穷远。后来，Patrice Le Calvez仅基于Brouwer理论和平面拓扑论证给出了该定理的不同证明[9]。这些方法允许通过证明指数为1的简单闭曲线的存在来改进结果。我们给出了这个定理的改进版本的一个新的、更简单的证明，并将其推广到无穷远处的无定向环
