$\mathbb{Z}_2$-Thurston norm in Sol manifolds and embeddability of non-orientable surfaces
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918406228279296 |
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| author | Du, Xiaoming Wang, Weibiao |
| author_facet | Du, Xiaoming Wang, Weibiao |
| contents | For every Sol manifold $M$, we determine the $\mathbb{Z}_2$-Thurston norm of every element in $H_2(M;\mathbb{Z}_2)$. Each Sol manifold is either a torus bundle over the circle or a torus semi-bundle, thus corresponds to a torus map. We discuss the action of this torus map on a curve complex for the torus, whose edges connect curve classes of intersection number 2. For torus bundles over the circle, the $\mathbb{Z}_2$-Thurston norm of any $\mathbb{Z}_2$-homology class equals either zero or the minimum translation distance under the action; and for torus semi-bundles, it equals either zero or the translation distance of a specific curve class. Moreover, we construct incompressible surfaces to realize all the $\mathbb{Z}_2$-homology classes.
As a consequence, for any torus bundle over the circle or torus semi-bundle, we determine which non-orientable closed surfaces can be embedded in it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_22894 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $\mathbb{Z}_2$-Thurston norm in Sol manifolds and embeddability of non-orientable surfaces Du, Xiaoming Wang, Weibiao Geometric Topology For every Sol manifold $M$, we determine the $\mathbb{Z}_2$-Thurston norm of every element in $H_2(M;\mathbb{Z}_2)$. Each Sol manifold is either a torus bundle over the circle or a torus semi-bundle, thus corresponds to a torus map. We discuss the action of this torus map on a curve complex for the torus, whose edges connect curve classes of intersection number 2. For torus bundles over the circle, the $\mathbb{Z}_2$-Thurston norm of any $\mathbb{Z}_2$-homology class equals either zero or the minimum translation distance under the action; and for torus semi-bundles, it equals either zero or the translation distance of a specific curve class. Moreover, we construct incompressible surfaces to realize all the $\mathbb{Z}_2$-homology classes. As a consequence, for any torus bundle over the circle or torus semi-bundle, we determine which non-orientable closed surfaces can be embedded in it. |
| title | $\mathbb{Z}_2$-Thurston norm in Sol manifolds and embeddability of non-orientable surfaces |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2603.22894 |