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

Abstract

We construct a map from the suspension G-spectrum Σ ∞ G 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-cobodism 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. CONTENTS 1. Introduction 2.1. Multicategories and multifunctors 2.2. Modules over rings on many objects as multifunctors 2.3. Equivalences of Waldhausen and Segal K-theory 2.4. Pseudo linear maps of modules 3. The map Σ ∞ G X + → A G (X) 3.1. Review of the definition of A G (X) 3.2. Homotopy discrete retractive spaces 3.3. Σ ∞ G X + as a spectral Mackey functor 3.4. Reconciling the different models for Σ ∞ G X 4. The fiber map of the map Σ ∞ G X + → A G (X) References M. Merling was supported by NSF grant DMS-1709461.