Monopole Floer homology and invariant theta characteristics
We describe a relationship between the monopole Floer homology of three‐manifolds and the geometry of Riemann surfaces. For an automorphism φ$\varphi$ of a compact Riemann surface Σ$\Sigma$ with quotient P1$\mathbb {P}^1$, there is a natural correspondence between theta characteristics L$L$ on Σ$\Si...
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| Published in: | Journal of the London Mathematical Society Vol. 109; no. 5 |
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| Format: | Journal Article |
| Language: | English |
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| ISSN: | 0024-6107, 1469-7750 |
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| Abstract | We describe a relationship between the monopole Floer homology of three‐manifolds and the geometry of Riemann surfaces. For an automorphism φ$\varphi$ of a compact Riemann surface Σ$\Sigma$ with quotient P1$\mathbb {P}^1$, there is a natural correspondence between theta characteristics L$L$ on Σ$\Sigma$ which are invariant under φ$\varphi$ and self‐conjugate spinc${\text{spin}}^c$ structures sL$\mathfrak {s}_L$ on the mapping torus Mφ$M_{\varphi }$ of φ$\varphi$. We show that the monopole Floer homology groups of (Mφ,sL)$(M_{\varphi },\mathfrak {s}_L)$ are explicitly determined by the eigenvalues of the (lift of the) action of φ$\varphi$ on H0(L)$H^0(L)$, the space of holomorphic sections of L$L$, and discuss several consequences of this description. Our result is based on a detailed analysis of the transversality properties of the Seiberg–Witten equations for suitable small perturbations. |
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| AbstractList | We describe a relationship between the monopole Floer homology of three‐manifolds and the geometry of Riemann surfaces. For an automorphism of a compact Riemann surface with quotient , there is a natural correspondence between theta characteristics on which are invariant under and self‐conjugate structures on the mapping torus of . We show that the monopole Floer homology groups of are explicitly determined by the eigenvalues of the (lift of the) action of on , the space of holomorphic sections of , and discuss several consequences of this description. Our result is based on a detailed analysis of the transversality properties of the Seiberg–Witten equations for suitable small perturbations. We describe a relationship between the monopole Floer homology of three‐manifolds and the geometry of Riemann surfaces. For an automorphism φ$\varphi$ of a compact Riemann surface Σ$\Sigma$ with quotient P1$\mathbb {P}^1$, there is a natural correspondence between theta characteristics L$L$ on Σ$\Sigma$ which are invariant under φ$\varphi$ and self‐conjugate spinc${\text{spin}}^c$ structures sL$\mathfrak {s}_L$ on the mapping torus Mφ$M_{\varphi }$ of φ$\varphi$. We show that the monopole Floer homology groups of (Mφ,sL)$(M_{\varphi },\mathfrak {s}_L)$ are explicitly determined by the eigenvalues of the (lift of the) action of φ$\varphi$ on H0(L)$H^0(L)$, the space of holomorphic sections of L$L$, and discuss several consequences of this description. Our result is based on a detailed analysis of the transversality properties of the Seiberg–Witten equations for suitable small perturbations. |
| Author | Lin, Francesco |
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| Snippet | We describe a relationship between the monopole Floer homology of three‐manifolds and the geometry of Riemann surfaces. For an automorphism φ$\varphi$ of a... We describe a relationship between the monopole Floer homology of three‐manifolds and the geometry of Riemann surfaces. For an automorphism of a compact... |
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