TITLE: formula for Eta invariant
QUESTION [18 upvotes]: Hirzebruch's signature formula is not valid for manifolds with boundary.
An error term is introduced by Atiyah-Patodi-Singer to fix it.More precisely:
  $$sign (M)=L(M)[M]+\eta(\partial M)$$
Yet better,in some situations,this invariant could be used to distinguish homeo types within a simple homotopy type.
My question is about basic properties of eta invariant:Is there a formula for 
$\bullet$ $\eta(M\times N)$ where $M$ and $N$ are closed manifolds
$\bullet$ $\eta(\widetilde{M})$ where $\widetilde{M}$ is a regular covering space of closed manifold $M$ 
$\bullet$ $\eta(2M)$ where $M$ is a manifold with boundary and $2M$ is the double $M\cup_{\partial M} M$.

REPLY [5 votes]: You may find some information from the following paper:
Donnelly, Harold
Eta invariant of a fibered manifold. 
Topology 15 (1976), no. 3, 247–252.
Review from Mathscinet