A Brouwer fixed-point theorem for graph endomorphisms

Abstract

We prove a Lefschetz formula L(T) = ∑x∈F i T(x) for graph endomorphisms T : G → G, where G is a general finite simple graph and F is the set of simplices fixed by T . The degree i T(x) of T at the simplex x is defined as (-1)dim(x) sign(T|x), a graded sign of the permutation of T restricted to the simplex. The Lefschetz number L(T) is defined similarly as in the continuum as L(T) =∑k(-1)k tr(Tk), where Tk is the map induced on the kth cohomology group Hk(G) of G. The theorem can be seen as a generalization of the Nowakowski-Rival fixed-edge theorem (Nowakowski and Rival in J. Graph Theory 3:339-350, 1979). A special case is the identity map T, where the formula reduces to the Euler-Poincaré formula equating the Euler characteristic with the cohomological Euler characteristic. The theorem assures that if L(T) is nonzero, then T has a fixed clique. A special case is the 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. If A is the automorphism group of a graph, we look at the average Lefschetz number L(G). We prove that this is the Euler characteristic of the chain G/A and especially an integer. We also show that as a consequence of the Lefschetz formula, the zeta function ζT(z) = exp(∑n=1∞ L(Tn) zn/n) is a product of two dynamical zeta functions and, therefore, has an analytic continuation as a rational function. This explicitly computable product formula involves the dimension and the signature of prime orbits. © 2013 Knill; licensee Springer.