Analytic surgery and the eta invariant

Abstract

Let (M, h) be an odd-dimensional compact spin manifold in whichH is an embedded hypersurface with quadratic defining functionx2∈C∞ (M). Let be the Dirac operator associated to the metric jhen where ∈>0 is a parameter. The limiting metric,g0, is an exact b-metric on the compact manifold with boundary\(\overline M \) obtained by cuttingM alongH and compactifying as a manifold with boundary, i.e. it givesM/H asymptotically cylindrical ends with cross-section\(\partial \overline M \), a double cover ofH. Under the assumption that the induced Dirac operator on this double cover is invertible we show that where is the ‘b’ version of the eta invariant introduced in [Me1],r1(∈) andr2(∈) are smooth, vanish at ∈=0 and are integrals of local geometric data, and where\(\tilde \eta ( \in )\) is the finite dimensional eta invariant, or signature, for the small eigenvalues of . If is invertible then\(\tilde \eta ( \in ) \equiv 0\) and Even if is not invertible this holds in ℝ/ℤ. In fact the discussion takes place more naturally in the context of the generalized Dirac operators associated to Hermitian Clifford modules.