Equivariant trisections for group actions on four-manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909469986783232 |
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| author | Meier, Jeffrey Scott, Evan |
| author_facet | Meier, Jeffrey Scott, Evan |
| contents | Let $G$ be a finite group, and let $X$ be a smooth, orientable, connected, closed 4-dimensional $G$-manifold.
Let $\mathcal{S}$ be a smooth, embedded, $G$-invariant surface in $X$.
We introduce the concept of a $G$-equivariant trisection of $X$ and the notion of $G$-equivariant bridge trisected position for $\mathcal{S}$ and establish that any such $X$ admits a $G$-equivariant trisection such that $\mathcal{S}$ is in equivariant bridge trisected position.
Our definitions are designed so that $G$-equivariant (bridge) trisections are determined by their spines; hence, the 4-dimensional equivariant topology of a $G$-manifold pair $(X,\mathcal{S})$ can be reduced to the 2-dimensional data of a $G$-equivariant shadow diagram.
As an application, we discuss how equivariant trisections can be used to study quotients of $G$-manifolds.
We also describe many examples of equivariant trisections, paying special attention to branched covering actions, hyperelliptic involutions, and linear actions on familiar manifolds such as $S^4$, $S^2\times S^2$, and $\mathbb{CP}^2$.
We show that equivariant trisections of genus at most one are geometric, and we give a partial classification for genus-two. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17999 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equivariant trisections for group actions on four-manifolds Meier, Jeffrey Scott, Evan Geometric Topology 57M60, 57M12, 57K40, 57K45, 57R65 Let $G$ be a finite group, and let $X$ be a smooth, orientable, connected, closed 4-dimensional $G$-manifold. Let $\mathcal{S}$ be a smooth, embedded, $G$-invariant surface in $X$. We introduce the concept of a $G$-equivariant trisection of $X$ and the notion of $G$-equivariant bridge trisected position for $\mathcal{S}$ and establish that any such $X$ admits a $G$-equivariant trisection such that $\mathcal{S}$ is in equivariant bridge trisected position. Our definitions are designed so that $G$-equivariant (bridge) trisections are determined by their spines; hence, the 4-dimensional equivariant topology of a $G$-manifold pair $(X,\mathcal{S})$ can be reduced to the 2-dimensional data of a $G$-equivariant shadow diagram. As an application, we discuss how equivariant trisections can be used to study quotients of $G$-manifolds. We also describe many examples of equivariant trisections, paying special attention to branched covering actions, hyperelliptic involutions, and linear actions on familiar manifolds such as $S^4$, $S^2\times S^2$, and $\mathbb{CP}^2$. We show that equivariant trisections of genus at most one are geometric, and we give a partial classification for genus-two. |
| title | Equivariant trisections for group actions on four-manifolds |
| topic | Geometric Topology 57M60, 57M12, 57K40, 57K45, 57R65 |
| url | https://arxiv.org/abs/2501.17999 |