The equivariant parametrized h-cobordism theorem, the non-manifold part

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We construct a map from the suspension G-spectrum ΣG∞M of a smooth compact G-manifold to the equivariant A-theory spectrum AG(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 reconcile 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.