An APS index theorem for even-dimensional manifolds with non-compact boundary

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 use this index to define the relative $\eta$-invariant $\eta (\mathcal{A}_1 , \mathcal{A}_0)$ of two strongly Callias-type operators, which are equal outside of a compact set. 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 $\eta (\mathcal{A}_1) - \eta (\mathcal{A}_0)$. We also define the spectral flow of a family of such operators and use it to compute the variation of the relative $\eta$-invariant.

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Maxim Braverman was partially supported by the Simons Foundation collaboration grant #G00005104.

Received 6 March 2018

Accepted 26 November 2018

Published 19 April 2021