Volume 11, issue 1 (2007)

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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 –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 EF(1) and we construct examples of degree d rational maps into S4. Moreover we explicitly give the moduli space of degree 1 rational maps into S4 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
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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