  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 (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 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.
