{"title":"单极浮子同源性和不变的 Theta 特性","authors":"Francesco Lin","doi":"10.1112/jlms.12895","DOIUrl":null,"url":null,"abstract":"<p>We describe a relationship between the monopole Floer homology of three-manifolds and the geometry of Riemann surfaces. For an automorphism <span></span><math>\n <semantics>\n <mi>φ</mi>\n <annotation>$\\varphi$</annotation>\n </semantics></math> of a compact Riemann surface <span></span><math>\n <semantics>\n <mi>Σ</mi>\n <annotation>$\\Sigma$</annotation>\n </semantics></math> with quotient <span></span><math>\n <semantics>\n <msup>\n <mi>P</mi>\n <mn>1</mn>\n </msup>\n <annotation>$\\mathbb {P}^1$</annotation>\n </semantics></math>, there is a natural correspondence between theta characteristics <span></span><math>\n <semantics>\n <mi>L</mi>\n <annotation>$L$</annotation>\n </semantics></math> on <span></span><math>\n <semantics>\n <mi>Σ</mi>\n <annotation>$\\Sigma$</annotation>\n </semantics></math> which are invariant under <span></span><math>\n <semantics>\n <mi>φ</mi>\n <annotation>$\\varphi$</annotation>\n </semantics></math> and self-conjugate <span></span><math>\n <semantics>\n <msup>\n <mtext>spin</mtext>\n <mi>c</mi>\n </msup>\n <annotation>${\\text{spin}}^c$</annotation>\n </semantics></math> structures <span></span><math>\n <semantics>\n <msub>\n <mi>s</mi>\n <mi>L</mi>\n </msub>\n <annotation>$\\mathfrak {s}_L$</annotation>\n </semantics></math> on the mapping torus <span></span><math>\n <semantics>\n <msub>\n <mi>M</mi>\n <mi>φ</mi>\n </msub>\n <annotation>$M_{\\varphi }$</annotation>\n </semantics></math> of <span></span><math>\n <semantics>\n <mi>φ</mi>\n <annotation>$\\varphi$</annotation>\n </semantics></math>. We show that the monopole Floer homology groups of <span></span><math>\n <semantics>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>M</mi>\n <mi>φ</mi>\n </msub>\n <mo>,</mo>\n <msub>\n <mi>s</mi>\n <mi>L</mi>\n </msub>\n <mo>)</mo>\n </mrow>\n <annotation>$(M_{\\varphi },\\mathfrak {s}_L)$</annotation>\n </semantics></math> are explicitly determined by the eigenvalues of the (lift of the) action of <span></span><math>\n <semantics>\n <mi>φ</mi>\n <annotation>$\\varphi$</annotation>\n </semantics></math> on <span></span><math>\n <semantics>\n <mrow>\n <msup>\n <mi>H</mi>\n <mn>0</mn>\n </msup>\n <mrow>\n <mo>(</mo>\n <mi>L</mi>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$H^0(L)$</annotation>\n </semantics></math>, the space of holomorphic sections of <span></span><math>\n <semantics>\n <mi>L</mi>\n <annotation>$L$</annotation>\n </semantics></math>, 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.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Monopole Floer homology and invariant theta characteristics\",\"authors\":\"Francesco Lin\",\"doi\":\"10.1112/jlms.12895\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We describe a relationship between the monopole Floer homology of three-manifolds and the geometry of Riemann surfaces. For an automorphism <span></span><math>\\n <semantics>\\n <mi>φ</mi>\\n <annotation>$\\\\varphi$</annotation>\\n </semantics></math> of a compact Riemann surface <span></span><math>\\n <semantics>\\n <mi>Σ</mi>\\n <annotation>$\\\\Sigma$</annotation>\\n </semantics></math> with quotient <span></span><math>\\n <semantics>\\n <msup>\\n <mi>P</mi>\\n <mn>1</mn>\\n </msup>\\n <annotation>$\\\\mathbb {P}^1$</annotation>\\n </semantics></math>, there is a natural correspondence between theta characteristics <span></span><math>\\n <semantics>\\n <mi>L</mi>\\n <annotation>$L$</annotation>\\n </semantics></math> on <span></span><math>\\n <semantics>\\n <mi>Σ</mi>\\n <annotation>$\\\\Sigma$</annotation>\\n </semantics></math> which are invariant under <span></span><math>\\n <semantics>\\n <mi>φ</mi>\\n <annotation>$\\\\varphi$</annotation>\\n </semantics></math> and self-conjugate <span></span><math>\\n <semantics>\\n <msup>\\n <mtext>spin</mtext>\\n <mi>c</mi>\\n </msup>\\n <annotation>${\\\\text{spin}}^c$</annotation>\\n </semantics></math> structures <span></span><math>\\n <semantics>\\n <msub>\\n <mi>s</mi>\\n <mi>L</mi>\\n </msub>\\n <annotation>$\\\\mathfrak {s}_L$</annotation>\\n </semantics></math> on the mapping torus <span></span><math>\\n <semantics>\\n <msub>\\n <mi>M</mi>\\n <mi>φ</mi>\\n </msub>\\n <annotation>$M_{\\\\varphi }$</annotation>\\n </semantics></math> of <span></span><math>\\n <semantics>\\n <mi>φ</mi>\\n <annotation>$\\\\varphi$</annotation>\\n </semantics></math>. We show that the monopole Floer homology groups of <span></span><math>\\n <semantics>\\n <mrow>\\n <mo>(</mo>\\n <msub>\\n <mi>M</mi>\\n <mi>φ</mi>\\n </msub>\\n <mo>,</mo>\\n <msub>\\n <mi>s</mi>\\n <mi>L</mi>\\n </msub>\\n <mo>)</mo>\\n </mrow>\\n <annotation>$(M_{\\\\varphi },\\\\mathfrak {s}_L)$</annotation>\\n </semantics></math> are explicitly determined by the eigenvalues of the (lift of the) action of <span></span><math>\\n <semantics>\\n <mi>φ</mi>\\n <annotation>$\\\\varphi$</annotation>\\n </semantics></math> on <span></span><math>\\n <semantics>\\n <mrow>\\n <msup>\\n <mi>H</mi>\\n <mn>0</mn>\\n </msup>\\n <mrow>\\n <mo>(</mo>\\n <mi>L</mi>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$H^0(L)$</annotation>\\n </semantics></math>, the space of holomorphic sections of <span></span><math>\\n <semantics>\\n <mi>L</mi>\\n <annotation>$L$</annotation>\\n </semantics></math>, 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.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-04-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12895\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12895","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们描述了三芒星的单极弗洛尔同源性与黎曼曲面几何之间的关系。对于具有商 P 1 $\mathbb {P}^1$ 的紧凑黎曼曲面 Σ $\Sigma$ 的自变形 φ $varphi$ 、在φ $\varphi$的映射环M φ $M_{varphi }$上,Σ $\Sigma$上在φ $\varphi$下不变的θ特性L $L$与自共轭自旋c ${text{spin}}^c$ 结构s L $\mathfrak {s}_L$ 之间存在自然的对应关系。我们证明了 ( M φ , s L ) $(M_{{varphi },\mathfrak {s}_L)$ 的单极弗洛尔同调群明确地由 φ $\varphi$ 对 H 0 ( L ) $H^0(L)$ 的(提升)作用的特征值决定,而 H 0 ( L ) $H^0(L)$ 是 L $L$ 的全形截面空间,并讨论了这种描述的若干后果。我们的结果基于对塞伯格-维滕方程在合适的小扰动下的横向性的详细分析。
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 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.
期刊介绍:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.