The Atiyah–Patodi–Singer Index on Manifolds with Non-compact Boundary

We study the index of the APS boundary value problem for a strongly Callias-type operator D on a complete Riemannian manifold M. We show that this index is equal to an index on a simpler manifold whose boundary is a disjoint union of two complete manifolds N and N₁. If the dimension of M is odd we show that the latter index depends only on the restrictions A and A₁ of D to N and N₁ and thus is an invariant of the boundary. We use this invariant to define the relative η-invariant η(A₁, A). We show that even though in our situation the η-invariants of A₁ and A are not defined, the relative η-invariant behaves as if it were the difference η(A₁) - η(A).