Lefschetz fixed point formula on a compact Riemannian manifold with boundary for some boundary conditions

TL;DR

This paper extends the Lefschetz fixed point formula to compact Riemannian manifolds with boundary using new boundary conditions and heat kernel methods, providing insights into fixed points of smooth maps.

Contribution

It introduces a Lefschetz fixed point formula on new de Rham complexes with boundary conditions, applying heat kernel techniques to manifolds with boundary.

Findings

Lefschetz fixed point formula derived for manifolds with boundary

Application of heat kernel method to boundary conditions

Analysis of fixed points near the boundary

Abstract

In [8] the authors introduced a pair of new de Rham complexes on a compact oriented Riemannian manifold with boundary by using a pair of new boundary conditions to discuss the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the Lefschetz fixed point formula on these complexes with respect to a smooth map having simple fixed points and satisfying some special condition near the boundary. For this purpose we are going to use the heat kernel method for the Lefschetz fixed point formula.