Volume 15, issue 4 (2011)

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Intersection theory of punctured pseudoholomorphic curves

Richard Siefring

Geometry & Topology 15 (2011) 2351–2457
Abstract

We study the intersection theory of punctured pseudoholomorphic curves in $4$–dimensional symplectic cobordisms. Using the asymptotic results of the author [Comm. Pure Appl. Math. 61(2008) 1631–84], we first study the local intersection properties of such curves at the punctures. We then use this to develop topological controls on the intersection number of two curves. We also prove an adjunction formula which gives a topological condition that will guarantee a curve in a given homotopy class is embedded, extending previous work of Hutchings [JEMS 4(2002) 313–61].

We then turn our attention to curves in the symplectization $ℝ×M$ of a $3$–manifold $M$ admitting a stable Hamiltonian structure. We investigate controls on intersections of the projections of curves to the $3$–manifold and we present conditions that will guarantee the projection of a curve to the $3$–manifold is an embedding.

Finally we consider an application concerning pseudoholomorphic curves in manifolds admitting a certain class of holomorphic open book decomposition and an application concerning the existence of generalized pseudoholomorphic curves, as introduced by Hofer [Geom. Func. Anal. (2000) 674–704] .

Keywords
pseudoholomorphic curves, symplectic field theory, Floer homology, intersection theory
Mathematical Subject Classification 2000
Primary: 32Q65
Secondary: 53D42, 57R58
Publication
Received: 4 June 2010
Revised: 19 June 2011
Accepted: 13 August 2011
Published: 25 December 2011
Proposed: Yasha Eliashberg
Seconded: Ronald Fintushel, Leonid Polterovich
Authors
 Richard Siefring Max Planck Institute for Math in Sci Inselstraße 22 D-04103 Leipzig Germany Department of Mathematics Michigan State University East Lansing, MI 48824 USA http://personal-homepages.mis.mpg.de/siefring/