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

Malkiewich, Cary; Merling, Mona

doi:10.1016/j.aim.2022.108242

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

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Malkiewich, Cary
Max Planck Institute for Mathematics, Max Planck Society;

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Merling, Mona
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1016/j.aim.2022.108242
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Citation

Malkiewich, C., & Merling, M. (2022). The equivariant parametrized h-cobordism theorem, the non-manifold part. Advances in Mathematics, 399: 108242. doi:10.1016/j.aim.2022.108242.

Cite as: https://hdl.handle.net/21.11116/0000-000A-2209-6

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.