ON THE GLUING PROBLEM FOR THE ETA-INVARIANT

We solve the gluing problem for the tl-invariant. Consider a generaliz ed Dirac operator D over a compact Riemannian manifold M that is parti tioned by a compact hypersurface N such that M := M(1) U-N M(2). We as sume that the Riemannian metric of M and D have a product structure ne ar N, i.e., D = I (partial derivative/partial derivative tau + D-N) wi th some Dime operator D-N On N. Using boundary conditions of Atiyah-Pa todi-Singer type parametrized by Lagrangian subspaces L(i) of kerD(N) we define selfadjoint extensions D-i, i = 1, 2, over M(i). We express the eta-invariant of D in terms of the eta-invariants of D-i ,an invar iant 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 neighborhood of N is long eno ugh, the vanishing of J is shown under certain regularity conditions. We apply this result in order to prove cutting and pasting formulas fo r the eta-invariant, a Wall nonadditivity result for the index of Atiy ah-Patodi-Singer boundary value problems and a splitting formula for t he spectral flow.