On the gluing problem for the \(\eta\)-invariant
Consider a generalized Dirac operator \(D\) over a compact Riemannian manifold \(M\) that is partitioned by a compact hypersurface \(N\) such that \(M := M_ 1 \cup_ N M_ 2\). Assume that the Riemannian metric of \(M\) and \(D\) have a product structure near \(N\), i.e. \(D = I(\partial/\partial r + D_ N)\) with some Dirac operator \(D_ N\) on \(N\). Using boundary conditions of Atiyah-Patodi-Singer-type parametrized by Lagrangian subspaces \(L_ i\) of \(\text{ker }D_ N\) one defines selfadjoint extensions \(D_ i\), \(i = 1,2\) over \(M_ i\). The main result is a formula expressing the \(\eta\)-invariant of \(D\) as a sum of the \(\eta\)- invariants of \(D_ i\), an invariant \(m(L_ 1,L_ 2)\) of the pair of the Lagrangian subspaces \(L_ 1\), \(L_ 2\) which is related to the Maslov index and an integer valued term \(J\). In the adiabatic limit, i.e. if a tubular neighbourhood of \(N\) is long enough, the vanishing of \(J\) is shown under certain regularity conditions. This result is applied in order to prove cutting and pasting formulas for the \(\eta\)-invariant, a Wall non-additivity result for the index of Atiyah-Patodi-Singer boundary value problems and a splitting formula for the spectral flow.
scientific article; zbMATH DE number 761169