ABSTRACT
This chapter analyses the geometric theory of η-invariants of Dirac operators on manifolds with boundary. One key geometric observation is that the exponentiated η-invariant naturally takes values in the determinant line of the boundary. As such it is intimately related to the geometry of determinant line bundles for families of Dirac operators. A family of Riemannian manifolds is a smooth fiber bundle π X → Z together with a metric on the relative tangent bundle T(X/Z) and a distribution of “horizontal” complements to T(X/Z) in TX. On a manifold with boundary elliptic boundary conditions are specified to obtain an operator with discrete spectrum. The boundary conditions introduced by Atiyah-Patodi-Singer are used. The chapter discusses the application of the geometry of the determinant line bundle. Suppose X → Z is a family of odd-dimensional manifolds with boundary. Then ∂X → Z is a family of closed even-dimensional manifolds. The determinant lines patch together to form a smooth determinant line bundle.
