The LefschetzHopf theorem and axioms for the Lefschetz number
Abstract
The reduced Lefschetz number, that is, the Lefschetz number minus 1, is proved to be the unique integervalued 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 (f_1, f_2, f_3) is a map of a cofiber sequence into itself, then L(f_2) = L(f_1) + L(f_3); (3) L(f) =  (degree(p_1 f e_1) + ... + degree(p_k f e_k)), where f is a map of a wedge of k circles, e_r is the inclusion of a circle into the rth summand and p_r 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 LefschetzHopf 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2004
 arXiv:
 arXiv:math/0403421
 Bibcode:
 2004math......3421A
 Keywords:

 Algebraic Topology;
 55M20
 EPrint:
 9 pages