#### Volume 11, issue 1 (2007)

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Pseudoholomorphic maps into folded symplectic four-manifolds

### Jens von Bergmann

Geometry & Topology 11 (2007) 1–45
 arXiv: math.SG/0511597
##### Abstract

Every oriented $4$–manifold admits a stable folded symplectic structure, which in turn determines a homotopy class of compatible almost complex structures that are discontinuous across the folding hypersurface (“fold”) in a controlled fashion. We define folded holomorphic maps, ie pseudoholomorphic maps that are discontinuous across the fold. The boundary values on the fold are mediated by tunneling maps which are punctured $\mathsc{ℋ}$–holomorphic maps into the folding hypersurface with prescribed asymptotics on closed characteristics.

Our main result is that the linearized operator of this boundary value problem is Fredholm, under the simplifying assumption that we have circle-invariant folds.

As examples we characterize the moduli space of maps into the folded elliptic fibration ${E}^{F}\left(1\right)$ and we construct examples of degree $d$ rational maps into ${S}^{4}$. Moreover we explicitly give the moduli space of degree 1 rational maps into ${S}^{4}$ and show that it possesses a natural compactification.

This aims to generalize the tools of holomorphic maps to all oriented $4$–manifolds by utilizing folded symplectic structures rather than other types of pre-symplectic structures as initiated by Taubes.

##### Keywords
pseudoholomorphic curves, boundary value problems on manifolds, folded symplectic structures
##### Mathematical Subject Classification 2000
Primary: 32Q65, 58J32
Secondary: 53C15, 57R17
##### Publication
Received: 11 January 2006
Accepted: 9 November 2006
Published: 23 February 2007
Proposed: Yasha Eliashberg
Seconded: Rob Kirby, Eleny Ionel
##### Authors
 Jens von Bergmann Department of Mathematics University of Notre Dame Notre Dame, IN 46556-4618