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

Abstract

We study the index of the APS boundary value problem for a strongly Callias-type operator $${\mathcal {D}}$$ 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 $$N_0$$ and $$N_1$$ . If the dimension of M is odd we show that the latter index depends only on the restrictions $${\mathcal {A}}_0$$ and $${\mathcal {A}}_1$$ of $${\mathcal {D}}$$ to $$N_0$$ and $$N_1$$ and thus is an invariant of the boundary. We use this invariant to define the relative $$\eta $$ -invariant $$\eta ({\mathcal {A}}_1,{\mathcal {A}}_0)$$ . We show that even though in our situation the $$\eta $$ -invariants of $${\mathcal {A}}_1$$ and $${\mathcal {A}}_0$$ are not defined, the relative $$\eta $$ -invariant behaves as if it were the difference $$\eta ({\mathcal {A}}_1)-\eta ({\mathcal {A}}_0)$$ .