## Geometry & Topology

### Embedded contact homology and Seiberg–Witten Floer cohomology I

Clifford Henry Taubes

#### Abstract

This is the first of five papers that construct an isomorphism between the embedded contact homology and Seiberg–Witten Floer cohomology of a compact $3$–manifold with a given contact $1$–form. This paper describes what is involved in the construction.

#### Article information

Source
Geom. Topol., Volume 14, Number 5 (2010), 2497-2581.

Dates
Accepted: 21 August 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732248

Digital Object Identifier
doi:10.2140/gt.2010.14.2497

Mathematical Reviews number (MathSciNet)
MR2746723

Zentralblatt MATH identifier
1275.57037

#### Citation

Taubes, Clifford Henry. Embedded contact homology and Seiberg–Witten Floer cohomology I. Geom. Topol. 14 (2010), no. 5, 2497--2581. doi:10.2140/gt.2010.14.2497. https://projecteuclid.org/euclid.gt/1513732248

#### References

• F Bourgeois, K Mohnke, Coherent orientations in symplectic field theory, Math. Z. 248 (2004) 123–146
• A Floer, Morse theory for fixed points of symplectic diffeomorphisms, Bull. Amer. Math. Soc. $($N.S.$)$ 16 (1987) 279–281
• A Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988) 513–547
• H Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three, Invent. Math. 114 (1993) 515–563
• H Hofer, Dynamics, topology, and holomorphic curves, from: “Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998)”, Extra Vol. I (1998) 255–280
• H Hofer, Holomorphic curves and dynamics in dimension three, from: “Symplectic geometry and topology (Park City, UT, 1997)”, (Y Eliashberg, L Traynor, editors), IAS/Park City Math. Ser. 7, Amer. Math. Soc. (1999) 35–101
• H Hofer, Holomorphic curves and real three-dimensional dynamics, Geom. Funct. Anal. (2000) 674–704 GAFA 2000 (Tel Aviv, 1999)
• 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
• H Hofer, K Wysocki, E Zehnder, Properties of pseudoholomorphic curves in symplectizations. III. Fredholm theory, from: “Topics in nonlinear analysis”, (J Escher, G Simonett, editors), Progr. Nonlinear Differential Equations Appl. 35, Birkhäuser, Basel (1999) 381–475
• M Hutchings, An index inequality for embedded pseudoholomorphic curves in symplectizations, J. Eur. Math. Soc. $($JEMS$)$ 4 (2002) 313–361
• M Hutchings, M Sullivan, Rounding corners of polygons and the embedded contact homology of $T\sp 3$, Geom. Topol. 10 (2006) 169–266
• M Hutchings, C H Taubes, Gluing pseudoholomorphic curves along branched covered cylinders. I, J. Symplectic Geom. 5 (2007) 43–137
• M Hutchings, C H Taubes, Gluing pseudoholomorphic curves along branched covered cylinders. II, J. Symplectic Geom. 7 (2009) 29–133
• A Jaffe, C Taubes, Vortices and monopoles. Structure of static gauge theories, Progress in Physics 2, Birkhäuser, Boston (1980)
• P Kronheimer, T Mrowka, Monopoles and three-manifolds, New Math. Monogr. 10, Cambridge Univ. Press (2007)
• C B Morrey, Jr, Multiple integrals in the calculus of variations, Grund. der math. Wissenschaften 130, Springer, New York (1966)
• D Quillen, Determinants of Cauchy–Riemann operators on Riemann surfaces, Funktsional. Anal. i Prilozhen. 19 (1985) 37–41, 96
• R Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders, Comm. Pure Appl. Math. 61 (2008) 1631–1684
• C H Taubes, Arbitrary $N$–vortex solutions to the first order Ginzburg–Landau equations, Comm. Math. Phys. 72 (1980) 277–292
• C H Taubes, The Seiberg–Witten and Gromov invariants, Math. Res. Lett. 2 (1995) 221–238
• C H Taubes, Seiberg–Witten and Gromov invariants for symplectic $4$–manifolds, (R Wentworth, editor), First Int. Press Lecture Ser. 2, Int. Press, Somerville, MA (2000)
• C H Taubes, Asymptotic spectral flow for Dirac operators, Comm. Anal. Geom. 15 (2007) 569–587
• C H Taubes, The Seiberg–Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007) 2117–2202
• C H Taubes, The Seiberg–Witten equations and the Weinstein conjecture. II. More closed integral curves of the Reeb vector field, Geom. Topol. 13 (2009) 1337–1417
• C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology II, Geom. Topol. 14 (2010) 2583–2720
• C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology III, Geom. Topol. 14 (2010) 2721–2817
• C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology IV, Geom. Topol. 14 (2010) 2819–2960
• C H Taubes, Embedded contact homology and Seiberg–Witten Floer cohomology V, Geom. Topol. 14 (2010) 2962–3000