\(SO(3)\)-invariants for 4-manifolds with \(b_ 2^ +=1\). II
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| Titel: | \(SO(3)\)-invariants for 4-manifolds with \(b_ 2^ +=1\). II |
|---|---|
| Autoren: | Kotschick, D., Morgan, J. W. |
| Verlagsinformationen: | International Press of Boston, Somerville, MA |
| Schlagwörter: | wall crossing formula, Applications of global analysis to structures on manifolds, anti-self-dual, one-dimensional positive definite subspace of the second homology group, self-dual harmonic 2-form, self-intersection form, \(\text{spin}^ c\)-structure, chamber, Donaldson polynomial invariant, ASD equation, gluing construction, Topology of the Euclidean \(4\)-space, \(4\)-manifolds |
| Beschreibung: | This paper extends the definition of Donaldson polynomial invariants to the case of manifolds with \(b_1 = 0\) and \(b^+_2 = 1\) for \(SO (3)\)-bundles where \(w_2\) lifts to an integral class. The paper generalizes earlier work of the first author [Part I: Proc. Lond. Math. Soc., III. Ser. 63, No. 2, 426-448 (1991; Zbl 0699.53036)]. It completes the proof that the values of the invariants only depend on the chamber containing the self-dual harmonic 2-form for the metric used to define the ASD equation, and establishes more general properties of the difference of values as the self dual 2-form crosses a wall. It establishes the conjecture made there that the value of an invariant on every chamber is determined by its value on any one chamber, and the invariant is defined for all chambers. Still unresolved is finding an explicit wall crossing formula as given by \textit{S. K. Donaldson} [J. Differ. Geom. 26, 141-168 (1987; Zbl 0631.57010)] for \(SU (2)\)-bundles with \(c_2 = 1\). It conjectures that there are systematic formulae for these difference terms involving only the classes defining the wall and the self-intersection form of the manifold. The results here fill a gap in the earlier paper of the first author cited above, but by an altogether different approach. The new argument uses a generalized gluing construction for gluing concentrated ASD connections over \(S^4\) into not necessarily ASD connections on \(M\). |
| Publikationsart: | Article |
| Dateibeschreibung: | application/xml |
| DOI: | 10.4310/jdg/1214454879 |
| Zugangs-URL: | https://zbmath.org/590837 |
| Dokumentencode: | edsair.c2b0b933574d..c6e97fdb4c5d826efa18f968b8a9725e |
| Datenbank: | OpenAIRE |
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| Items | – Name: Title Label: Title Group: Ti Data: \(SO(3)\)-invariants for 4-manifolds with \(b_ 2^ +=1\). II – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Kotschick%2C+D%2E%22">Kotschick, D.</searchLink><br /><searchLink fieldCode="AR" term="%22Morgan%2C+J%2E+W%2E%22">Morgan, J. W.</searchLink> – Name: Publisher Label: Publisher Information Group: PubInfo Data: International Press of Boston, Somerville, MA – Name: Subject Label: Subject Terms Group: Su Data: <searchLink fieldCode="DE" term="%22wall+crossing+formula%22">wall crossing formula</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="%22anti-self-dual%22">anti-self-dual</searchLink><br /><searchLink fieldCode="DE" term="%22one-dimensional+positive+definite+subspace+of+the+second+homology+group%22">one-dimensional positive definite subspace of the second homology group</searchLink><br /><searchLink fieldCode="DE" term="%22self-dual+harmonic+2-form%22">self-dual harmonic 2-form</searchLink><br /><searchLink fieldCode="DE" term="%22self-intersection+form%22">self-intersection form</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="%22chamber%22">chamber</searchLink><br /><searchLink fieldCode="DE" term="%22Donaldson+polynomial+invariant%22">Donaldson polynomial invariant</searchLink><br /><searchLink fieldCode="DE" term="%22ASD+equation%22">ASD equation</searchLink><br /><searchLink fieldCode="DE" term="%22gluing+construction%22">gluing construction</searchLink><br /><searchLink fieldCode="DE" term="%22Topology+of+the+Euclidean+%5C%284%5C%29-space%2C+%5C%284%5C%29-manifolds%22">Topology of the Euclidean \(4\)-space, \(4\)-manifolds</searchLink> – Name: Abstract Label: Description Group: Ab Data: This paper extends the definition of Donaldson polynomial invariants to the case of manifolds with \(b_1 = 0\) and \(b^+_2 = 1\) for \(SO (3)\)-bundles where \(w_2\) lifts to an integral class. The paper generalizes earlier work of the first author [Part I: Proc. Lond. Math. Soc., III. Ser. 63, No. 2, 426-448 (1991; Zbl 0699.53036)]. It completes the proof that the values of the invariants only depend on the chamber containing the self-dual harmonic 2-form for the metric used to define the ASD equation, and establishes more general properties of the difference of values as the self dual 2-form crosses a wall. It establishes the conjecture made there that the value of an invariant on every chamber is determined by its value on any one chamber, and the invariant is defined for all chambers. Still unresolved is finding an explicit wall crossing formula as given by \textit{S. K. Donaldson} [J. Differ. Geom. 26, 141-168 (1987; Zbl 0631.57010)] for \(SU (2)\)-bundles with \(c_2 = 1\). It conjectures that there are systematic formulae for these difference terms involving only the classes defining the wall and the self-intersection form of the manifold. The results here fill a gap in the earlier paper of the first author cited above, but by an altogether different approach. The new argument uses a generalized gluing construction for gluing concentrated ASD connections over \(S^4\) into not necessarily ASD connections on \(M\). – Name: TypeDocument Label: Document Type Group: TypDoc Data: Article – Name: Format Label: File Description Group: SrcInfo Data: application/xml – Name: DOI Label: DOI Group: ID Data: 10.4310/jdg/1214454879 – Name: URL Label: Access URL Group: URL Data: <link linkTarget="URL" linkTerm="https://zbmath.org/590837" linkWindow="_blank">https://zbmath.org/590837</link> – Name: AN Label: Accession Number Group: ID Data: edsair.c2b0b933574d..c6e97fdb4c5d826efa18f968b8a9725e |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.4310/jdg/1214454879 Languages: – Text: Undetermined Subjects: – SubjectFull: wall crossing formula Type: general – SubjectFull: Applications of global analysis to structures on manifolds Type: general – SubjectFull: anti-self-dual Type: general – SubjectFull: one-dimensional positive definite subspace of the second homology group Type: general – SubjectFull: self-dual harmonic 2-form Type: general – SubjectFull: self-intersection form Type: general – SubjectFull: \(\text{spin}^ c\)-structure Type: general – SubjectFull: chamber Type: general – SubjectFull: Donaldson polynomial invariant Type: general – SubjectFull: ASD equation Type: general – SubjectFull: gluing construction Type: general – SubjectFull: Topology of the Euclidean \(4\)-space, \(4\)-manifolds Type: general Titles: – TitleFull: \(SO(3)\)-invariants for 4-manifolds with \(b_ 2^ +=1\). II Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Kotschick, D. – PersonEntity: Name: NameFull: Morgan, J. W. IsPartOfRelationships: – BibEntity: Identifiers: – Type: issn-locals Value: edsair |
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