^{1}Department of Mathematics, Dartmouth College, Hanover 03755-1890, NH, USA ^{2}Department of Mathematics, University of California, Los Angeles 90095-1555, CA, USA

The reduced Lefschetz number, that is, L(⋅)−1 where
L(⋅) denotes the Lefschetz number, is proved to be the
unique integer-valued function λ on self-maps of compact
polyhedra which is constant on homotopy classes such that (1)
λ(fg)=λ(gf) for f:X→Y and g:Y→X; (2) if (f1,f2,f3) is a map of a cofiber
sequence into itself, then λ(f1)=λ(f1)+λ(f3); (3) λ(f)=−(deg(p1fe1)+⋯+deg(pkfek)), where f is a self-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 self-map of a polyhedron and I(f) is
the fixed point index of f on all of X, then we show that
I(⋅)−1 satisfies the above axioms. This gives a new
proof of the normalization theorem: if f:X→X is a
self-map of a polyhedron, then I(f) equals the Lefschetz number
L(f) of f. This result is equivalent to the Lefschetz-Hopf
theorem: if f:X→X is a self-map 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.