## Tokyo Journal of Mathematics

### Lefschetz Theory, Geometric Thom Forms and the Far Point Set

#### Abstract

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 $L(f) \neq \chi(M)$ is infinite unless $M$ is a sphere. There are homology classes supported near $\text{Far}(f)$ which determine $L(f)-\chi(M).$ 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 $f=\text{Id}.$ 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 $f$.

#### Article information

Source
Tokyo J. Math., Volume 27, Number 2 (2004), 337-355.

Dates
First available in Project Euclid: 5 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1244208393

Digital Object Identifier
doi:10.3836/tjm/1244208393

Mathematical Reviews number (MathSciNet)
MR2107507

Zentralblatt MATH identifier
1158.53325

#### Citation

FRUMOSU, Mihail; ROSENBERG, Steven. Lefschetz Theory, Geometric Thom Forms and the Far Point Set. Tokyo J. Math. 27 (2004), no. 2, 337--355. doi:10.3836/tjm/1244208393. https://projecteuclid.org/euclid.tjm/1244208393

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