# Atiyah-Patodi-Singer for manifolds with cusps

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).

• This is a nice question, but probably very hard. A starting point might be a version of the eta-invariant for the Dirac operator (rather than signature operator) on an odd-dimensional hyperbolic manifold with a cusp, due to J. Park (Amer. J. Math. 127 (2005), no. 3, 493–534 or arxiv.org/abs/math/0111175.) Aug 5 '18 at 18:28
• Perhaps, the following will help "Index theorems on manifolds with straight ends" by Werner Ballmann, Jochen Bruning and Gilles Carron, mathematik.hu-berlin.de/~geomanal/bruening_publications/…. This is not on my fingertips so I cannot claim anything definite. Aug 5 '18 at 19:53
• @DannyRuberman: Thanks, Danny. Ok, I see: it might be hard, and I'm no real differential geometer. However, this question appears extremely interesting in the context of geometric cobordisms of hyperbolic 3-manifolds. If you're interested, I can write more by e-mail. Aug 5 '18 at 21:18
• @IgorBelegradek: Thanks, Igor! Just in case, my question is related to this paper (arxiv.org/pdf/math/0007197.pdf) of Long and Reid about geometric cobordisms of hyperbolic 3-manifolds. Aug 5 '18 at 21:19

This type of questions has been investigated systematically by Melrose in the framework of 'c-calculus', where $c$ stands for the cusp. The basic idea, if I recall correctly is to blow up the boundary with the cusp three times (two blow ups for the boundary, an extra blow-up for the cusp on the boundary) and work with the heat operator associated with the Dirac operator on the blow-up space. Normally this does not help you "gain" anything since the metric is arbitrary. In your case, however the metric is product type. So your can explicitly recover the $\eta$-invariant and there will be an extra contribution from the third blow up. The exact meaning of this contribution may be difficult to interpret from the view of spectral flow.
I am not entirely sure what is a good reference for this, though. There is no good reference for $c$-calculus (Melrose has a book on $b$-calculus, which is dense). This paper may be the closest thing I can find at the moment (there should be expository articles on this written by Rafe, here is an article by Paul). I am no longer working in the area, so there may be mistakes from my poor memory.