Constructing symplectic forms on 4–manifolds which vanish on circles

Gay, David T; Kirby, Robion

doi:10.2140/gt.2004.8.743

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Constructing symplectic forms on 4–manifolds which vanish on circles

David T Gay and Robion Kirby

Geometry & Topology 8 (2004) 743–777

DOI: 10.2140/gt.2004.8.743

arXiv: math.GT/0401186

Abstract

Given a smooth, closed, oriented 4–manifold $X$ and $α \in H_{2} (X, ℤ)$ such that $α \cdot α > 0$ , a closed $2$ –form is constructed, Poincaré dual to $α$ , which is symplectic on the complement of a finite set of unknotted circles $Z$ . The number of circles, counted with sign, is given by $d = (c_{1} {(s)}^{2} - 3 σ (X) - 2 χ (X)) ∕ 4$ , where $s$ is a certain ${spin}^{ℂ}$ structure naturally associated to $ω$ .

Keywords

symplectic, $4$–manifold, $\mathrm{spin}^C$, almost complex, harmonic

Mathematical Subject Classification 2000

Primary: 57R17

Secondary: 57M50, 32Q60

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Publication

Received: 17 January 2004

Revised: 6 May 2004

Accepted: 16 May 2004

Published: 18 May 2004

Proposed: Yasha Eliashberg

Seconded: Leonid Polterovich, Simon Donaldson

Authors

David T Gay
CIRGET Université du Québec à Montréal Case Postale 8888 Succursale centre-ville Montréal Quebec H3C 3P8 Canada

Robion Kirby
Department of Mathematics University of California Berkeley California 94720 USA

Volume 8, issue 2 (2004)