A model for the assembly map of bordism-invariant functors
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914046702256128 |
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| author | Levin, Jordan Nocera, Guglielmo Saunier, Victor |
| author_facet | Levin, Jordan Nocera, Guglielmo Saunier, Victor |
| contents | We study oplax colimits of stable categories, of hermitian categories and of Poincaré categories in nice cases. This allows us to produce a categorical model of the assembly map of a bordism-invariant functor of Poincaré 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 Poincaré categories proved by Calmès-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 Poincaré-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05238 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A model for the assembly map of bordism-invariant functors Levin, Jordan Nocera, Guglielmo Saunier, Victor K-Theory and Homology Algebraic Topology Category Theory Geometric Topology 18F25, 19D35, 18N65 We study oplax colimits of stable categories, of hermitian categories and of Poincaré categories in nice cases. This allows us to produce a categorical model of the assembly map of a bordism-invariant functor of Poincaré 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 Poincaré categories proved by Calmès-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 Poincaré-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. |
| title | A model for the assembly map of bordism-invariant functors |
| topic | K-Theory and Homology Algebraic Topology Category Theory Geometric Topology 18F25, 19D35, 18N65 |
| url | https://arxiv.org/abs/2506.05238 |