Let $(M,g\_M)$ be a closed oriented Riemannian manifold that has a fibration structure
$$
Y \rightarrow M \overset{\pi}{\rightarrow} B
$$
where $(Y,g\_Y)$ and $(B,g\_B)$ are closed Riemannian manifolds such that $\pi$ is a Riemannian submersion.


Now we define $g\_{\epsilon}=\epsilon^{-2}\pi^\*g\_B+g\_Y$ and $\tilde g\_{\epsilon}=g\_{\epsilon}+\alpha\_{\epsilon}$ for some error term $\alpha\_{\epsilon}=O(\epsilon^{\tau-2})$ for some $\tau>0$. If we denote the signature operators on $M$ with respect to $g\_{\epsilon}$ and $\tilde g\_{\epsilon}$ by $A\_{\epsilon}$ and $\tilde A\_{\epsilon}$ respectively. Can we conclude that


$$
\lim\_{\epsilon \to 0} \eta(A\_{\epsilon})= \lim\_{\epsilon \to 0} \eta(\tilde A\_{\epsilon}),
$$
if the former exists?



Perturbation of the adiabatic limit
I think the answer is "no" in general (whenever the dimensions are such that $\eta(A\_\epsilon)\ne 0$) because you can use $\alpha\_\epsilon$ to change
the fibre-wise metric $g\_Y$ in any way you like.


So take the special case where $\dim Y\equiv 3$ mod $4$ and $\dim B\equiv 0$
mod $4$, and assume that $B$ has nonzero signature. Changing the fibre-wise metric also changes the $\eta$-invariant of the fibre, which is the degree $0$ contribution to the $\eta$-form, and hence contributes to the adiabatic limit formula.


