Brouwer’s theorem follows from Lefschetz because a manifold M homeomorphic to the unit ball has H k ( M ) and is trivial for k > 0 so that L ( T ) = 1 assures the existence of a fixed point. The third chapter of and provides more history. In 1928, Hopf extended this to arbitrary finite Euclidean simplicial complexes and proved that if T has no fixed point then L ( T ) = 0. More general than Brouwer is Lefschetz’ fixed-point theorem ∑ x ∈ F i T ( x ) = L ( T ) from 1926, which assures that if the Lefschetz number L ( T ) of a continuous transformation on a manifold is nonzero, then T has a fixed point. It is also useful for the theorem of Perron-Frobenius in linear algebra, which is one of the mathematical foundations for the page rank used to measure the relevance of nodes in a network. It has its use for example in game theory: the Kakutani generalization has been used to prove Nash equilibria. First tackled by Poincaré in 1887 and by Bohl in 1904 in the context of differential equations, then by Hadamard in 1910 and Brouwer in 1912, in general, it is now a basic application in algebraic topology. MSC:58J20, 47H10, 37C25, 05C80, 05C82, 05C10, 90B15, 57M15, 55M20.īrouwer’s fixed-point theorem assures that any continuous transformation on the closed ball in Euclidean space has a fixed point. This explicitly computable product formula involves the dimension and the signature of prime orbits. We also show that as a consequence of the Lefschetz formula, the zeta function ζ T ( z ) = exp ( ∑ n = 1 ∞ L ( T n ) z n n ) is a product of two dynamical zeta functions and, therefore, has an analytic continuation as a rational function. We prove that this is the Euler characteristic of the chain G / A and especially an integer. ![]() If A is the automorphism group of a graph, we look at the average Lefschetz number L ( G ). 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. The theorem assures that if L ( T ) is nonzero, then T has a fixed clique. 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 can be seen as a generalization of the Nowakowski-Rival fixed-edge theorem (Nowakowski and Rival in J. The Lefschetz number L ( T ) is defined similarly as in the continuum as L ( T ) = ∑ k ( − 1 ) k tr ( T k ), where T k is the map induced on the k th cohomology group H k ( G ) of G. 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. ![]() 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.
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