Dirac operators, heat kernels and microlocal analysis Part II: analytic surgery

Abstract

Let X be a closed Riemannian manifold and let H\hookrightarrow X be an embedded hypersurface. Let X=X_+ \cup_H X_- be a decomposition of X into two manifolds with boundary, with X_+ \cap X_- = H. In this expository article, surgery -- or gluing -- formul\ae for several geometric and spectral invariants associated to a Dirac-type operator \eth_X on X are presented. Considered in detail are: the index of \eth_X, the index bundle and the determinant bundle associated to a family of such operators, the eta invariant and the analytic torsion. In each case the precise form of the surgery theorems, as well as the different techniques used to prove them, are surveyed.