Dear Colleagues and Friends,

Please let me know if you are aware of any references to the following question.

The classical result of Atiyah, Patodi and Singer tells us that if $W$ is a compact oriented Riemannian 4-manifold with boundary $M$ and, moreover, if we assume that near M the metric is isometric to a product, then $$ sign(W)= \frac{1}{3} \int_W p_1 - \eta(M),$$ where $p_1$ is the differential form representing the first Pontryagin class of $W$, and $\eta$ is the eta-invariant of $M$.

What about the case when both $W$ and $M$ are hyperbolic manifolds and are allowed to have cusps? Or, say, $W$ and $M$ are Riemannian as above, with infinite ends of finite volume, on which the metric is isometric to a product? (which will be the case if both are hyperbolic with cusps - I'm sure that this is not a very general setting :-))

Any information will be appreciated. Please excuse my ignorance as differential geometer.

Dear Colleagues and Friends,

Please let me know if you are aware of any references to the following question.

The classical result of Atiyah, Patodi and Singer tells us that if $W$ is a compact oriented Riemannian 4-manifold with boundary $M$ and, moreover, if we assume that near M the metric is isometric to a product, then $$ sign(W)= \frac{1}{3} \int_W p_1 - \eta(M),$$ where $p_1$ is the differential form representing the first Pontryagin class of $W$, and $\eta$ is the eta-invariant of $M$.

What about the case when both $W$ and $M$ are hyperbolic manifolds and are allowed to have cusps? Or, say, $W$ and $M$ are Riemannian as above, with infinite ends of finite volume, on which the metric is isometric to a product?

Any information will be appreciated. Please excuse my ignorance as differential geometer.

Correction: for hyperbolic manifolds with boundary and/or cusps the metric near the boundary is not a product (otherwise we won't have finite volume, I'd suppose). There is a correction term (already in the paper by Long and Reid), which is, however, vanishing due to various reasons (e.g. for totally geodesic boundary its second fundamental form vanishes, and it annihilates the correction term, and for the cusp case we can deduce it from the fact that the volume of cusp section by a small horoball is 0 in the limit, and we integrate over that horoball).