In a previous work of the authors [“The refined analytic torsion and a well-posed boundary condition for the odd signature operator”, Preprint, arXiv:1004.1753], de Rham complexes \(\{\Omega^\bullet_{\widetilde{P}_\alpha}(M),d\}\), \(\alpha\in\{0,1\}\) are defined for a compact oriented manifold \(M\) with boundary \(\partial M\). They also showed that, if \(q\equiv \alpha\mod{2}\), then \(H^q\{\Omega^\bullet_{\widetilde{P}_\alpha}(M),d\}\) is isomorphic to \(H^q(M,\partial M)\); otherwise, \(H^q\{\Omega^\bullet_{\widetilde{P}_\alpha}(M),d\}\) is isomorphic to \(H^q(M)\). Here \(H^*\{\Omega^\bullet_{\widetilde{P}_\alpha}(M),d\}\) denotes the cohomology of the complex \(\{\Omega^\bullet_{\widetilde{P}_\alpha}(M),d\}\), and \(H^*(M)\) and \(H^*(M,\partial M)\), respectively, denote the de Rham cohomology and its relative version of \(M\) and \((M,\partial M)\).
In the present article, the authors derive, using the heat kernel associated with the Laplacian of the complex \(\{\Omega^\bullet_{\widetilde{P}_\alpha}(M),d\}\), a fixed point formula for the Lefschetz number of a smooth boundary-preserving map \(f: M\to M\), under the condition that \(f\) has only simple fixed points and is a local isometry on the boundary.