#### Volume 3, issue 1 (2003)

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Grafting Seiberg–Witten monopoles

### Stanislav Jabuka

Algebraic & Geometric Topology 3 (2003) 155–185
 arXiv: math.SG/0110285
##### Abstract

We demonstrate that the operation of taking disjoint unions of $J$–holomorphic curves (and thus obtaining new $J$–holomorphic curves) has a Seiberg–Witten counterpart. The main theorem asserts that, given two solutions $\left({A}_{i},{\psi }_{i}\right)$, $i=0,1$ of the Seiberg–Witten equations for the ${Spin}^{c}$–structures ${W}_{{E}_{i}}^{+}={E}_{i}\oplus \left({E}_{i}\otimes {K}^{-1}\right)$ (with certain restrictions), there is a solution $\left(A,\psi \right)$ of the Seiberg–Witten equations for the ${Spin}^{c}$–structure ${W}_{E}$ with $E={E}_{0}\otimes {E}_{1}$, obtained by “grafting” the two solutions $\left({A}_{i},{\psi }_{i}\right)$.

##### Keywords
symplectic 4–manifolds, Seiberg–Witten gauge theory, $J$–holomorphic curves
##### Mathematical Subject Classification 2000
Primary: 53D99, 57R57
Secondary: 53C27, 58J05
##### Publication
Received: 24 November 2002
Revised: 27 January 2003
Accepted: 13 February 2003
Published: 21 February 2003
##### Authors
 Stanislav Jabuka Department of Mathematics Columbia University 2990 Broadway New York, NY 10027 USA