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arXiv · 2001.05563

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

Cary Malkiewich·Mona Merling
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Abstract

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.

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Cary Malkiewich, Mona Merling. 2020-01-15. The equivariant parametrized $h$-cobordism theorem, the non-manifold part. https://arxiv.org/abs/2001.05563

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