Amenable-signature vanishing for integral solvable bordisms

The supplied extending coefficient map $\psi :{\pi }_{1}W\to G$ satisfies $\psi (({\pi }_{1}W{)}^{(n+1)})\subset G^{(n+1)}=1$, by induction on commutators. Hence it factors through the height-$(n+1)$ quotient carrying the isotropic lifted classes in [F1]. The actual smooth $W$ has its smooth boundary collar and is in particular an admissible collared topological manifold for [F2]. Its integral homology, finite pair lists and unit dual pairings are exactly [F1]; its meridian has infinite image by the Statement. Thus every hypothesis of [F2], including $n=0$ and $R={\mathbb{F}}_2$ when applicable, is satisfied by this same $W$ and extending map.

Applying [F2] gives ${\mathrm{sign}}_G^{(2)}(W,\psi )=0=\mathrm{sign}(W)$. This invokes its actual weak-exactness and finite-projective calculation: no closed range, finite CW type of a TOP manifold, or lift of the height-$n$ dual classes to the height-$(n+1)$ coefficient system is inferred. The latter calculation uses only the lifted isotropic classes and their ordinary unit duals, precisely the supplied data checked in step 1.1.

Since this very bound is smooth, [F3] identifies its actual defect with $\rho (M(K),\varphi )$ under the Definition's smooth bounding convention. Therefore $\rho (M(K),\varphi )={\mathrm{sign}}_G^{(2)}(W,\psi )-\mathrm{sign}(W)=0$. The proof covers arbitrary target groups by [F2]'s same-image induction and [F3]'s target-enlargement independence. No comparison with a nonsmooth TOP bound or an analytic eta definition is required for this original smooth-solution Statement.