Perturbation of the adiabatic limit

Question

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?

Answer 1

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.