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We give new axioms for the Lefschetz number based on Hadwiger'scharacterization of the Euler characteristic as the unique lattice valuation onpolyhedra which takes value 1 on simplices. In the setting of maps on abstractsimplicial complexes, we show that the Lefschetz number is unique with respectto a valuation axiom and an axiom specifying the value on a simplex. Theseaxioms lead naturally to the classical computation of the Lefschetz number as atrace in homology. We then extend this approach to continuous maps ofpolyhedra, assuming an extra homotopy invariance axiom. We also show that thishomotopy axiom can be weakened.

Axioms for the Lefschetz number as a lattice valuation