#### Volume 8, issue 2 (2004)

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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
 arXiv: math.GT/0401186
##### Abstract

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

##### Keywords
symplectic, $4$–manifold, $\mathrm{spin}^C$, almost complex, harmonic
##### Mathematical Subject Classification 2000
Primary: 57R17
Secondary: 57M50, 32Q60