#### Geometry & Topology, Vol. 8 (2004)
Paper no. 20, pages 743--777.

## Constructing symplectic forms on 4-manifolds which vanish on circles

### David T Gay, Robion Kirby

**Abstract**.
Given a smooth, closed, oriented 4-manifold X and alpha in H_2(X,Z)
such that alpha.alpha > 0, a closed 2-form w is constructed, Poincare
dual to alpha, which is symplectic on the complement of a finite set
of unknotted circles. The number of circles, counted with sign, is
given by d = (c_1(s)^2 -3sigma(X) -2chi(X))/4, where s is a certain
spin^C structure naturally associated to w.
**Keywords**.
Symplectic, 4-manifold, spin^C, almost complex, harmonic

**AMS subject classification**.
Primary: 57R17.
Secondary: 57M50, 32Q60.

**DOI:** 10.2140/gt.2004.8.743

**E-print:** `arXiv:math.GT/0401186`

Submitted to GT on 17 January 2004.
(Revised 6 May 2004.)
Paper accepted 16 May 2004.
Paper published 18 May 2004.

Notes on file formats
David T Gay, Robion Kirby

CIRGET, Universite du Quebec a Montreal, Case Postale 8888

Succursale centre-ville, Montreal (QC) H3C 3P8, Canada

and

Department of Mathematics, University of California

Berkeley, CA 94720, USA

Email: gay@math.uqam.ca, kirby@math.berkeley.edu

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