\(\mathbb Z_p\)-equivariant Spin\(^c\)-structures
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| Title: | \(\mathbb Z_p\)-equivariant Spin\(^c\)-structures |
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| Authors: | Cho, Yong Seung, Hong, Yoon Hi |
| Publisher Information: | Korean Mathematical Society, Seoul |
| Subject Terms: | 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.) |
| Description: | 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 \(\chi(\Sigma)+\Sigma\cdot\Sigma=0\). |
| Document Type: | Article |
| File Description: | application/xml |
| DOI: | 10.4134/bkms.2003.40.1.017 |
| Access URL: | https://zbmath.org/1990038 |
| Accession Number: | edsair.c2b0b933574d..88b26d4c3e0daf4e180cd94e4ce9b15e |
| Database: | OpenAIRE |
Nájsť tento článok vo Web of Science | Abstract: | 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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| DOI: | 10.4134/bkms.2003.40.1.017 |