2004/03/24 by Martin Arkowitz, Arkowitz, Martin, Robert F. Brown +1 · 1 citation
Computer Science · Mathematics · #55M20 #Algebraic Topology (math.AT) #FOS: Mathematics #Topological and Geometric Data Analysis #math.AT #msc:55M20
paper · pdf · doi:10.48550/arxiv.math/0403421
9 pages
arxiv created 2004/03/24 · openalex publication_date 2004/03/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The reduced Lefschetz number, that is, the Lefschetz number minus 1, is proved to be the unique integer-valued function L on selfmaps of compact polyhedra which is constant on homotopy classes such that (1) L(fg) = L(gf), for f:X -->Y and g:Y -->X; (2) if (f1, f2, f3) is a map of a cofiber sequence into itself, then L(f2) = L(f1) + L(f3); (3) L(f) = - (degree(p1 f e1) + ... + degree(pk f ek)), where f is a map of a wedge of k circles, er is the inclusion of a circle into the rth summand and pr is the projection onto the rth summand. If f:X -->X is a selfmap of a polyhedron and I(f) is the fixed point index of f on all of X, then we show that I minus 1 satisfies the above axioms. This gives a new proof of the Normalization Theorem: If f:X -->X is a selfmap of a polyhedron, then I(f) equals the Lefschetz number of f. This result is equivalent to the Lefschetz-Hopf Theorem: If f: X -->X is a selfmap of a finite simplicial complex with a finite number of fixed points, each lying in a maximal simplex, then the Lefschetz number of f is the sum of the indices of all the fixed points of f.