A linearization map for genuine equivariant algebraic $K$-theory
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866915762518622208 |
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| author | Calle, Maxine Chan, David Mejia, Andres |
| author_facet | Calle, Maxine Chan, David Mejia, Andres |
| contents | We introduce a version of algebraic $K$-theory for coefficient systems of rings which is valued in genuine $G$-spectra for a finite group $G$. We use this construction to build a genuine $G$-spectrum $K_G(\mathbb{Z}[\underline{π_1(X)}])$ associated to a $G$-space $X$, which provides a home for equivariant versions of classical invariants like the Wall finiteness obstruction and Whitehead torsion. We provide a comparison between our $K$-theory spectrum and the equivariant $A$-theory of Malkiewich--Merling via a genuine equivariant linearization map. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_08025 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A linearization map for genuine equivariant algebraic $K$-theory Calle, Maxine Chan, David Mejia, Andres Algebraic Topology K-Theory and Homology 55P91 (Primary) 19D10, 19L47 (Secondary) We introduce a version of algebraic $K$-theory for coefficient systems of rings which is valued in genuine $G$-spectra for a finite group $G$. We use this construction to build a genuine $G$-spectrum $K_G(\mathbb{Z}[\underline{π_1(X)}])$ associated to a $G$-space $X$, which provides a home for equivariant versions of classical invariants like the Wall finiteness obstruction and Whitehead torsion. We provide a comparison between our $K$-theory spectrum and the equivariant $A$-theory of Malkiewich--Merling via a genuine equivariant linearization map. |
| title | A linearization map for genuine equivariant algebraic $K$-theory |
| topic | Algebraic Topology K-Theory and Homology 55P91 (Primary) 19D10, 19L47 (Secondary) |
| url | https://arxiv.org/abs/2309.08025 |