## Geometry & Topology

### Pseudoholomorphic punctured spheres in $\mathbb{R}{\times}(S^{1}{\times}S^{2})$: Properties and existence

Clifford Henry Taubes

#### Abstract

This is the first of at least two articles that describe the moduli spaces of pseudoholomorphic, multiply punctured spheres in $ℝ×(S1×S2)$ as defined by a certain natural pair of almost complex structure and symplectic form. This article proves that all moduli space components are smooth manifolds. Necessary and sufficient conditions are also given for a collection of closed curves in $S1×S2$ to appear as the set of $|s|→∞$ limits of the constant $s∈ℝ$ slices of a pseudoholomorphic, multiply punctured sphere.

#### Article information

Source
Geom. Topol., Volume 10, Number 2 (2006), 785-928.

Dates
Accepted: 9 May 2006
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513799743

Digital Object Identifier
doi:10.2140/gt.2006.10.785

Mathematical Reviews number (MathSciNet)
MR2240906

Zentralblatt MATH identifier
1134.53045

#### Citation

Taubes, Clifford Henry. Pseudoholomorphic punctured spheres in $\mathbb{R}{\times}(S^{1}{\times}S^{2})$: Properties and existence. Geom. Topol. 10 (2006), no. 2, 785--928. doi:10.2140/gt.2006.10.785. https://projecteuclid.org/euclid.gt/1513799743

#### References

• F Bourgeois, Y Eliashberg, H Hofer, K Wysocki, E Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003) 799–888
• Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000) 560–673
• D T Gay, R Kirby, Constructing symplectic forms on 4–manifolds which vanish on circles, Geom. Topol. 8 (2004) 743–777
• H Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993) 515–563
• H H W Hofer, Dynamics, topology, and holomorphic curves, Doc. Math. (1998) 255–280
• H Hofer, Holomorphic curves and dynamics in dimension three, from: “Symplectic geometry and topology (Park City, UT, 1997)”, IAS/Park City Math. Ser. 7, Amer. Math. Soc., Providence, RI (1999) 35–101
• H Hofer, K Wysocki, E Zehnder, Properties of pseudo-holomorphic curves in symplectisations. II. Embedding controls and algebraic invariants, Geom. Funct. Anal. 5 (1995) 270–328
• H Hofer, K Wysocki, E Zehnder, Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996) 337–379 See [HWZ1-corr?] for correction
• H Hofer, K Wysocki, E Zehnder, Correction to: “Properties of pseudoholomorphic curves in symplectisations. I. Asymptotics”, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998) 535–538
• H Hofer, K Wysocki, E Zehnder, Properties of pseudoholomorphic curves in symplectizations. III. Fredholm theory, from: “Topics in nonlinear analysis”, Progr. Nonlinear Differential Equations Appl. 35, Birkhäuser, Basel (1999) 381–475
• K Honda, An openness theorem for harmonic 2–forms on 4–manifolds, Illinois J. Math. 44 (2000) 479–495
• M Hutchings, M Sullivan, The periodic Floer homology of a Dehn twist, Algebr. Geom. Topol. 5 (2005) 301–354 (electronic)
• D McDuff, The local behaviour of holomorphic curves in almost complex $4$–manifolds, J. Differential Geom. 34 (1991) 143–164
• D McDuff, D Salamon, $J$–holomorphic curves and quantum cohomology, University Lecture Series 6, American Mathematical Society, Providence, RI (1994)
• C H Taubes, The geometry of the Seiberg–Witten invariants, Doc. Math. (1998) 493–504
• C H Taubes, The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on $S\sp 1\times B\sp 3$, Geom. Topol. 2 (1998) 221–332
• C H Taubes, Seiberg–Witten invariants, self-dual harmonic 2–forms and the Hofer–Wysocki–Zehnder formalism, from: “Surveys in differential geometry”, Surv. Differ. Geom., VII, Int. Press, Somerville, MA (2000) 625–672
• C H Taubes, A compendium of pseudoholomorphic beasts in $\Bbb R\times (S\sp 1\times S\sp 2)$, Geom. Topol. 6 (2002) 657–814 (electronic)
• R Ye, }, Trans. Amer. Math. Soc. 342 (1994) 671–694