Lifting relations in right orderable groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915075565027328 |
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| author | Linton, Marco |
| author_facet | Linton, Marco |
| contents | In this article we study the following problem: given a chain complex $A_*$ of free $\mathbb{Z}G$-modules, when is $A_*$ isomorphic to the cellular chain complex of some simply connected $G$-CW-complex? Such a chain complex is called realisable. Wall studied this problem in the 60's and reduced it to a problem involving only the second differential $d_2$, now known as the relation lifting problem. We show that if $G$ is right orderable and $d_2$ is given by a matrix of a certain form, then $A_*$ is realisable. As a special case, we solve the relation lifting problem for right orderable groups with cyclic relation module. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_17057 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lifting relations in right orderable groups Linton, Marco Group Theory Algebraic Topology 20F05 (Primary), 20F60, 20J05 (Secondary) In this article we study the following problem: given a chain complex $A_*$ of free $\mathbb{Z}G$-modules, when is $A_*$ isomorphic to the cellular chain complex of some simply connected $G$-CW-complex? Such a chain complex is called realisable. Wall studied this problem in the 60's and reduced it to a problem involving only the second differential $d_2$, now known as the relation lifting problem. We show that if $G$ is right orderable and $d_2$ is given by a matrix of a certain form, then $A_*$ is realisable. As a special case, we solve the relation lifting problem for right orderable groups with cyclic relation module. |
| title | Lifting relations in right orderable groups |
| topic | Group Theory Algebraic Topology 20F05 (Primary), 20F60, 20J05 (Secondary) |
| url | https://arxiv.org/abs/2412.17057 |