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The far point set of a self-map of a closed Riemannian manifold M is defined to be the set of points mapped into their cut locus. We prove that the far point set of a map f with Lefschetz number L(f ) = χ(M) is infinite unless M is a sphere. There are homology classes supported near Far(f ) which determine L(f ) − χ(M). Using geometric representatives of Thom classes, we obtain a geometric integral formula for the the Lefschetz number, which specializes to the Chern-Gauss-Bonnet formula when fdoi:10.3836/tjm/1244208393 fatcat:mie7dpbp2vhh5h3vtarj3jby4e