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\(\mathbb Z_p\)-equivariant Spin\(^c\)-structures

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Názov: \(\mathbb Z_p\)-equivariant Spin\(^c\)-structures
Autori: Cho, Yong Seung, Hong, Yoon Hi
Informácie o vydavateľovi: Korean Mathematical Society, Seoul
Predmety: General low-dimensional topology, Applications of global analysis to structures on manifolds, Equivariant algebraic topology of manifolds, involutions, 4-manifold with a \(\mathbb{Z}_p\)-action, equivariant, Finite transformation groups, moduli space, \(\text{Spin}^c\)-structure, Seiberg-Witten equations, Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
Popis: Let \(X\) be a closed, oriented, Riemannian \(4\)-manifold with a \(\mathbb Z_p\)-action \(\sigma\) and a \(\text{Spin}^c\)-structure \(\tilde P\) equivariant with respect to some lift of \(\sigma\). The authors consider Seiberg-Witten equations, define an invariant moduli space and calculate its virtual dimension. As an application to orientation preserving involutions, the authors show that if, in addition to above assumptions, the manifold \(X\) is of simple type, \(b_2^+(X)>1\), the set of fixed points \(\Sigma\) is an oriented, connected, compact \(2\)-dimensional submanifold, \(\Sigma\cdot\Sigma\geq 0\), \([\Sigma]\neq 0\in H_2(X,\mathbb Z)\), \(SW(\tilde P)\neq 0\), and \(c_1(L)[\Sigma]=0\), where \(L\) is the determinant line bundle associated with \(\tilde P\), then \(\chi(\Sigma)+\Sigma\cdot\Sigma=0\).
Druh dokumentu: Article
Popis súboru: application/xml
DOI: 10.4134/bkms.2003.40.1.017
Prístupová URL adresa: https://zbmath.org/1990038
Prístupové číslo: edsair.c2b0b933574d..88b26d4c3e0daf4e180cd94e4ce9b15e
Databáza: OpenAIRE
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  Data: \(\mathbb Z_p\)-equivariant Spin\(^c\)-structures
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  Data: <searchLink fieldCode="AR" term="%22Cho%2C+Yong+Seung%22">Cho, Yong Seung</searchLink><br /><searchLink fieldCode="AR" term="%22Hong%2C+Yoon+Hi%22">Hong, Yoon Hi</searchLink>
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  Data: Korean Mathematical Society, Seoul
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  Data: <searchLink fieldCode="DE" term="%22General+low-dimensional+topology%22">General low-dimensional topology</searchLink><br /><searchLink fieldCode="DE" term="%22Applications+of+global+analysis+to+structures+on+manifolds%22">Applications of global analysis to structures on manifolds</searchLink><br /><searchLink fieldCode="DE" term="%22Equivariant+algebraic+topology+of+manifolds%22">Equivariant algebraic topology of manifolds</searchLink><br /><searchLink fieldCode="DE" term="%22involutions%22">involutions</searchLink><br /><searchLink fieldCode="DE" term="%224-manifold+with+a+%5C%28%5Cmathbb{Z}%5Fp%5C%29-action%22">4-manifold with a \(\mathbb{Z}_p\)-action</searchLink><br /><searchLink fieldCode="DE" term="%22equivariant%22">equivariant</searchLink><br /><searchLink fieldCode="DE" term="%22Finite+transformation+groups%22">Finite transformation groups</searchLink><br /><searchLink fieldCode="DE" term="%22moduli+space%22">moduli space</searchLink><br /><searchLink fieldCode="DE" term="%22%5C%28%5Ctext{Spin}^c%5C%29-structure%22">\(\text{Spin}^c\)-structure</searchLink><br /><searchLink fieldCode="DE" term="%22Seiberg-Witten+equations%22">Seiberg-Witten equations</searchLink><br /><searchLink fieldCode="DE" term="%22Specialized+structures+on+manifolds+%28spin+manifolds%2C+framed+manifolds%2C+etc%2E%29%22">Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)</searchLink>
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  Data: Let \(X\) be a closed, oriented, Riemannian \(4\)-manifold with a \(\mathbb Z_p\)-action \(\sigma\) and a \(\text{Spin}^c\)-structure \(\tilde P\) equivariant with respect to some lift of \(\sigma\). The authors consider Seiberg-Witten equations, define an invariant moduli space and calculate its virtual dimension. As an application to orientation preserving involutions, the authors show that if, in addition to above assumptions, the manifold \(X\) is of simple type, \(b_2^+(X)>1\), the set of fixed points \(\Sigma\) is an oriented, connected, compact \(2\)-dimensional submanifold, \(\Sigma\cdot\Sigma\geq 0\), \([\Sigma]\neq 0\in H_2(X,\mathbb Z)\), \(SW(\tilde P)\neq 0\), and \(c_1(L)[\Sigma]=0\), where \(L\) is the determinant line bundle associated with \(\tilde P\), then \(\chi(\Sigma)+\Sigma\cdot\Sigma=0\).
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  Data: 10.4134/bkms.2003.40.1.017
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        Value: 10.4134/bkms.2003.40.1.017
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    Subjects:
      – SubjectFull: General low-dimensional topology
        Type: general
      – SubjectFull: Applications of global analysis to structures on manifolds
        Type: general
      – SubjectFull: Equivariant algebraic topology of manifolds
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      – SubjectFull: involutions
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      – SubjectFull: 4-manifold with a \(\mathbb{Z}_p\)-action
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      – SubjectFull: equivariant
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      – SubjectFull: Finite transformation groups
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      – SubjectFull: moduli space
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      – SubjectFull: \(\text{Spin}^c\)-structure
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      – SubjectFull: Seiberg-Witten equations
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      – SubjectFull: Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
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      – TitleFull: \(\mathbb Z_p\)-equivariant Spin\(^c\)-structures
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