Von Neumann index theorems for manifolds with boundary
Some extensions of the notions involved in the well-known Atiyah-Singer index theorem have been given by Atiyah (the definition of the real- valued index of the elliptic differential operator on infinite Galois coverings of closed manifolds, equivariant with respect to the Galois group, by using the von Neumann generalized dimension), by Cheeger-Gromov (the generalized eta invariants for the infinite Galois coverings), by the author and G. Peric (the eta invariants for foliations). In the present paper the author proves the following results concerning the geometric Dirac operators and the index theorem for foliations. Theorem 1.1. Let \(D\) be a Dirac operator on a compact manifold \(M\) with boundary, acting on the sections of a graded Clifford module \(S\), with grading operator \(\varepsilon\). Assume that the data \((D,S,\varepsilon)\) have a product structure near the boundary. Let \(\widetilde{M}\) be a Galois covering with Galois group \(\Gamma\). Let \((\widetilde{D},\widetilde{S},\widetilde{\varepsilon})\) be the lift of the data \((D,S,\varepsilon)\) to \(\widetilde{M}\). Let \(\widetilde{B}\) be the Atiyah-Patodi-Singer boundary condition associated to \(\widetilde{D}\). Then the \(\Gamma\) index of the Breuer-Fredholm operator \(\widetilde{D}\) with boundary condition \(\widetilde{B}\) is given by \[ \text{ind}_ \Gamma(\widetilde{D}) = \int_ M\text{ch}(\sigma_ D)Td(M) - {\eta_ \Gamma(0) + h\over 2}, \] where \(\eta_ \Gamma(0)\) is the \(\Gamma\)-eta invariant on the boundary of \(\widetilde{M}\). Theorem 1.2. Let \((M,{\mathcal F})\) be a compact foliated manifold with boundary, with the foliation \(\mathcal F\) transverse to the boundary. Let \(D_{\mathcal F}\) be a leafwise Dirac operator acting leafwise on a Clifford module \(S\) over the tangent bundle \(T{\mathcal F}\) to the foliation. Assume that \(S\) is graded with grading operator \(\varepsilon\). Let \(\nu\) be a holonomy invariant transverse measure for the foliation \(\mathcal F\). Further assume that the data \((D_{\mathcal F},S,\varepsilon)\) have a product structure near the boundary. Let \(B_{\mathcal F}\) be the family of Atiyah- Patodi-Singer boundary conditions corresponding to the family \(D_{\mathcal F}\). Then the \(\nu\)-index of this family of Breuer-Fredholm operators is finite and \[ \text{ind}_ \nu(D_{\mathcal F}) = \langle \text{ch}(\sigma_{D_{\mathcal F}})Td(M),\nu\rangle- {\eta_ \nu(0) + h\over 2} \] where \(\eta_ \nu(0)\) is the foliation eta invariant.
scientific article; zbMATH DE number 488226