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arXiv · 2506.05238

A model for the assembly map of bordism-invariant functors

Jordan Levin·Guglielmo Nocera·Victor Saunier
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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.

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Jordan Levin, Guglielmo Nocera, Victor Saunier. 2025-06-05. A model for the assembly map of bordism-invariant functors. https://arxiv.org/abs/2506.05238

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