Higher Spherical Scissors Congruence I: Hopf Algebra
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912599675764736 |
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| author | Klang, Inbar Kuijper, Josefien Malkiewich, Cary Mehrle, David Wittich, Thor |
| author_facet | Klang, Inbar Kuijper, Josefien Malkiewich, Cary Mehrle, David Wittich, Thor |
| contents | In the study of the generalization of Hilbert's Third Problem to spherical geometry, Sah constructed a Hopf algebra of spherical polytopes with product given by join and coproduct given by a generalized Dehn invariant. Using Zakharevich's reinterpretation of scissors congruence via algebraic K-theory, we lift the Sah algebra to an $(E_\infty, E_1)$-Hopf algebra spectrum whose $π_0$ is the classical Sah algebra. As an application, we show that the reduced spherical scissors congruence $K$-theory groups $\widetilde K_{2n}\big(\mathcal{P}^{S^{2k+1}}_{O(2k+2)}\big)$ are nonzero for all nonnegative integers $n$ and $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_18009 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Higher Spherical Scissors Congruence I: Hopf Algebra Klang, Inbar Kuijper, Josefien Malkiewich, Cary Mehrle, David Wittich, Thor K-Theory and Homology Algebraic Topology In the study of the generalization of Hilbert's Third Problem to spherical geometry, Sah constructed a Hopf algebra of spherical polytopes with product given by join and coproduct given by a generalized Dehn invariant. Using Zakharevich's reinterpretation of scissors congruence via algebraic K-theory, we lift the Sah algebra to an $(E_\infty, E_1)$-Hopf algebra spectrum whose $π_0$ is the classical Sah algebra. As an application, we show that the reduced spherical scissors congruence $K$-theory groups $\widetilde K_{2n}\big(\mathcal{P}^{S^{2k+1}}_{O(2k+2)}\big)$ are nonzero for all nonnegative integers $n$ and $k$. |
| title | Higher Spherical Scissors Congruence I: Hopf Algebra |
| topic | K-Theory and Homology Algebraic Topology |
| url | https://arxiv.org/abs/2509.18009 |