  We construct a map from the suspension $G$-spectrum $\Sigma_G^\infty M$ of a
smooth compact $G$-manifold to the equivariant $A$-theory spectrum $A_G(M)$,
and we show that its fiber is, on fixed points, a wedge of stable $h$-cobordism
spectra. This map is constructed as a map of spectral Mackey functors, which is
compatible with tom Dieck style splitting formulas on fixed points. In order to
synthesize different definitions of the suspension $G$-spectrum as a spectral
Mackey functor, we present a new perspective on spectral Mackey functors,
viewing them as multifunctors on indexing categories for "rings on many
objects" and modules over such. This perspective should be of independent
interest.
