A model for the assembly map of bordism-invariant functors

TL;DR

This paper develops a categorical model for the assembly map of bordism-invariant functors in Poincaré categories, extending previous splitting results to include twists and generalizing known decompositions.

Contribution

It introduces a new categorical framework for the assembly map in bordism-invariant functors, enabling the extension of Shaneson splitting to twisted cases and broader invariants.

Findings

Categorical model of the assembly map with explicit kernel

Generalization of Shaneson splitting to twisted settings

Application to twisted Bass-Heller-Swan decompositions

Abstract

We study oplax colimits of stable categories, of hermitian categories and of Poincar\'e categories in nice cases. This allows us to produce a categorical model of the assembly map of a bordism-invariant functor of Poincar\'e categories which is also a Verdier projection, whose kernel we explicitly describe. As a direct application, we generalize the Shaneson splitting for bordism-invariant functors of Poincar\'e categories proved by Calm\`es-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle to allow for twists. We also show our methods can tackle their general twisted Shaneson splitting of Poincar\'e-Verdier localizing invariants which specifies to a twisted Bass-Heller-Swan decomposition for the underlying stable categories, generalizing part of recent work of Kirstein-Kremer.