Geometric invariants of locally compact groups: the homological perspective
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| Format: | Preprint |
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2024
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| _version_ | 1866913968830808064 |
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| author | Bux, Kai-Uwe Hartmann, Elisa Quintanilha, José Pedro |
| author_facet | Bux, Kai-Uwe Hartmann, Elisa Quintanilha, José Pedro |
| contents | In this paper we develop the homological version of $Σ$-theory for locally compact Hausdorff groups, leaving the homotopical version for another paper. Both versions are connected by a Hurewicz-like theorem. They can be thought of as directional versions of type $\mathrm{CP}_m$ and type $\mathrm{C}_m$, respectively. And classical $Σ$-theory is recovered if we equip an abstract group with the discrete topology. This paper provides criteria for type $\mathrm{CP}_m$ and homological locally compact $Σ^m$. Given a short exact sequence with kernel of type $\mathrm{CP}_m$, we can derive $Σ^m$ of the extension on the sphere that vanishes on the kernel from the quotient and likewise. Given a short exact sequence with abelian quotient, $Σ$-theory on the extension can tell if the kernel is of type $\mathrm{CP}_m$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_13272 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric invariants of locally compact groups: the homological perspective Bux, Kai-Uwe Hartmann, Elisa Quintanilha, José Pedro Algebraic Topology Group Theory Metric Geometry In this paper we develop the homological version of $Σ$-theory for locally compact Hausdorff groups, leaving the homotopical version for another paper. Both versions are connected by a Hurewicz-like theorem. They can be thought of as directional versions of type $\mathrm{CP}_m$ and type $\mathrm{C}_m$, respectively. And classical $Σ$-theory is recovered if we equip an abstract group with the discrete topology. This paper provides criteria for type $\mathrm{CP}_m$ and homological locally compact $Σ^m$. Given a short exact sequence with kernel of type $\mathrm{CP}_m$, we can derive $Σ^m$ of the extension on the sphere that vanishes on the kernel from the quotient and likewise. Given a short exact sequence with abelian quotient, $Σ$-theory on the extension can tell if the kernel is of type $\mathrm{CP}_m$. |
| title | Geometric invariants of locally compact groups: the homological perspective |
| topic | Algebraic Topology Group Theory Metric Geometry |
| url | https://arxiv.org/abs/2411.13272 |