Tokyo Journal of Mathematics

Lefschetz Theory, Geometric Thom Forms and the Far Point Set


Full-text: Open access


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

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

First available in Project Euclid: 5 June 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


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.

Export citation


  • N. Berline, E. Getzler and M. Vergne, Heat Kernels and Dirac Operators, Springer (1992).
  • J. M. Bismut and W. Zhang, An Extension of a Theorem of Cheeger and Müller, Astérisque 205 (1992) Soc. Math. France.
  • R. Bott and L. Tu, Differential Forms in Algebraic Topology, Springer (1982).
  • C. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. Ec. Norm. Super. 13 (1980), 419–435.
  • M. Frumosu, Mathai-Quillen formalism and Lefschetz theory, Ph.D. thesis, Boston University (1997).
  • S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Springer (1987).
  • P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, Publish or Perish (1984).
  • V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall (1974).
  • F. B. Harvey and H. B. Lawson, Jr., A theory of characteristic currents associated with a singular connection, Astérisque 213 (1993), 1–268.
  • P. Li, On the Sobolev constant and the p-spectrum of a compact Riemannian manifold, Ann. Sci. Ec. Norm. Super. 13 (1980), 451–469.
  • V. Mathai and D. Quillen, Superconnections, Thom classes, and equivariant differential forms, Topology 25 (1986), 85–110.