Lefschetz Theory, Geometric Thom Forms and the Far Point Set

2004 Tokyo Journal of Mathematics  
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 f
more » ... net formula when f = Id. We compute this formula explicitly for constant curvature metrics. Finally, we give upper and lower bounds for L(f ) in terms of the geometry and topology of M and the differential of f .
doi:10.3836/tjm/1244208393 fatcat:mie7dpbp2vhh5h3vtarj3jby4e