On the gluing problem for the η-invariant

Abstract

We solve the gluing problem for the η-invariant. Consider a generalized Dirac operator D over a compact Riemannian manifold M that is partitioned by a compact hypersurface N such that M:- M1∪NM2. We assume that the Riemannian metric of M and D have a product structure near N, i.e., D = I(d/dτ + DN) with some Dirac operator DNon N. Using boundary conditions of Atiyah-Patodi-Singer type parametrized by Lagrangian subspaces Liof ker DNwe define selfadjoint extensions Di, i = 1, 2, over Mi. We express the η-invariant of D in terms of the η-invariants of Di, an invariant m(L1, L2) of the pair of the Lagrangian subspaces L1, L2, which is related to the Maslov index and an integer-valued term J. In the adiabatic limit, i.e., if a tubular neighborhood of N is long enough, the vanishing of J is shown under certain regularity conditions. We apply this result in order to prove cutting and pasting formulas for the η-invariant, a Wall nonadditivity result for the index of Atiyah-Patodi-Singer boundary value problems and a splitting formula for the spectral flow. © 1995, International Press of Boston, Inc. All Rights Reserved.