## Geometry & Topology

### Intersection theory of punctured pseudoholomorphic curves

Richard Siefring

#### 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] .

#### Article information

Source
Geom. Topol., Volume 15, Number 4 (2011), 2351-2457.

Dates
Revised: 19 June 2011
Accepted: 13 August 2011
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732370

Digital Object Identifier
doi:10.2140/gt.2011.15.2351

Mathematical Reviews number (MathSciNet)
MR2862160

Zentralblatt MATH identifier
1246.32028

Subjects
Primary: 32Q65: Pseudoholomorphic curves
Secondary: 53D42: Symplectic field theory; contact homology 57R58: Floer homology

#### Citation

Siefring, Richard. Intersection theory of punctured pseudoholomorphic curves. Geom. Topol. 15 (2011), no. 4, 2351--2457. doi:10.2140/gt.2011.15.2351. https://projecteuclid.org/euclid.gt/1513732370

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