Mayer-Vietoris formula for determinants of elliptic operators of Laplace-Beltrami type. (After Burghelea, Friedlander and Kappeler) (Q2752843)

Let \(A\) be an elliptic self-adjoint positive definite operator of Laplace type defined on a closed oriented Riemannian manifold \((M,g)\). Let \(\zeta(s):=Tr_{L^2}A^{-s}\) be the zeta function and let \(\log\det(A)=-\partial_s\zeta(s)|_{s=0}\) be the regularized functional determinant. Let \(M_\Gamma\) be the compact manifold with boundary defined by cutting \(M\) open along an oriented submanifold \(\Gamma\) of codimension \(1\). Let \(A_{\Gamma,B}\) be the realization of \(A\) on \(M_\Gamma\) with Dirichlet boundary conditions and let \(R\) be the Dirichlet to Neumann operator defined over \(\Gamma\). The author gives a `short cut presentation' of the Mayer-Vietoris formula due to \textit{D. Burghelea, L. Friedlander} and \textit{T. Kappeler} [J. Funct. Anal. 107, No. 1, 34-65 (1992; Zbl 0759.58043); see also Commun. Math. Phys. 138, 1-18 (1991; Zbl 0734.58043) and the erratum in Commun. Math. Phys. 150, No. 2, 431 (1992; Zbl 0764.58032)]: NEWLINE\[NEWLINE\det(A)=c\cdot\det(A_{\Gamma,B})\det(R)NEWLINE\]NEWLINE where \(c\) is locally computable in terms of local invariants on \(\Gamma\); if \(\dim M\) is even, then \(c=1\).NEWLINENEWLINEFor the entire collection see [Zbl 0953.00035].