2013/07/08 by P. Christopher Staecker, Staecker, P. Christopher
Computer Science · Mathematics · #52B45 #55M20 #Advanced Combinatorial Mathematics #Algebraic Topology (math.AT) #FOS: Mathematics #General Topology (math.GN) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.AT #math.GN #msc:52B45 #msc:55M20
paper · pdf · doi:10.48550/arxiv.1307.2131
8 pages
arxiv created 2013/07/08 · openalex publication_date 2013/07/08 · arxiv updated 2013/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give new axioms for the Lefschetz number based on Hadwiger's characterization of the Euler characteristic as the unique lattice valuation on polyhedra which takes value 1 on simplices. In the setting of maps on abstract simplicial complexes, we show that the Lefschetz number is unique with respect to a valuation axiom and an axiom specifying the value on a simplex. These axioms lead naturally to the classical computation of the Lefschetz number as a trace in homology. We then extend this approach to continuous maps of polyhedra, assuming an extra homotopy invariance axiom. We also show that this homotopy axiom can be weakened.