AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Sutured ECH is a natural invariant
About this Title
Çağatay Kutluhan, Steven Sivek and C. H. Taubes
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 275, Number 1350
ISBNs: 978-1-4704-5054-0 (print); 978-1-4704-7016-6 (online)
DOI: https://doi.org/10.1090/memo/1350
Published electronically: December 22, 2021
Table of Contents
Chapters
- 1. Introduction
- 2. Sutured ECH and some related constructions
- 3. Independence of the almost complex structure
- 4. Independence of the contact form
- 5. Some properties of the contact class
- 6. Invariance under gluing 1-handles
- 7. Stabilization and a canonical version of sutured ECH
- A. Appendix by C. H. Taubes
Abstract
We show that sutured embedded contact homology is a natural invariant of sutured contact $3$-manifolds which can potentially detect some of the topology of the space of contact structures on a $3$-manifold with boundary. The appendix, by C. H. Taubes, proves a compactness result for the completion of a sutured contact $3$-manifold in the context of Seiberg–Witten Floer homology, which enables us to complete the proof of naturality.- John A. Baldwin and Steven Sivek, Naturality in sutured monopole and instanton homology, J. Differential Geom. 100 (2015), no. 3, 395–480. MR 3352794
- F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003), 799–888. MR 2026549, DOI 10.2140/gt.2003.7.799
- V. Colin, P. Ghiggini, and K. Honda. Embedded contact homology and open book decompositions. 2013, math.SG/1008.2734v2.
- Vincent Colin, Paolo Ghiggini, Ko Honda, and Michael Hutchings, Sutures and contact homology I, Geom. Topol. 15 (2011), no. 3, 1749–1842. MR 2851076, DOI 10.2140/gt.2011.15.1749
- Fan Ding and Hansjörg Geiges, The diffeotopy group of $S^1\times S^2$ via contact topology, Compos. Math. 146 (2010), no. 4, 1096–1112. MR 2660686, DOI 10.1112/S0010437X09004606
- Y. Eliashberg, Classification of overtwisted contact structures on $3$-manifolds, Invent. Math. 98 (1989), no. 3, 623–637. MR 1022310, DOI 10.1007/BF01393840
- Yakov Eliashberg, Contact $3$-manifolds twenty years since J. Martinet’s work, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 1-2, 165–192 (English, with French summary). MR 1162559
- Hansjörg Geiges and Mirko Klukas, The fundamental group of the space of contact structures on the 3-torus, Math. Res. Lett. 21 (2014), no. 6, 1257–1262. MR 3335846, DOI 10.4310/MRL.2014.v21.n6.a3
- Emmanuel Giroux, Convexité en topologie de contact, Comment. Math. Helv. 66 (1991), no. 4, 637–677 (French). MR 1129802, DOI 10.1007/BF02566670
- G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 2d ed. MR 0046395
- Michael Hutchings, Embedded contact homology and its applications, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 1022–1041. MR 2827830
- Michael Hutchings, Lecture notes on embedded contact homology, Contact and symplectic topology, Bolyai Soc. Math. Stud., vol. 26, János Bolyai Math. Soc., Budapest, 2014, pp. 389–484. MR 3220947, DOI 10.1007/978-3-319-02036-5_{9}
- Michael Hutchings and Clifford Henry Taubes, Gluing pseudoholomorphic curves along branched covered cylinders. II, J. Symplectic Geom. 7 (2009), no. 1, 29–133. MR 2491716
- Michael Hutchings and Clifford Henry Taubes, Proof of the Arnold chord conjecture in three dimensions, II, Geom. Topol. 17 (2013), no. 5, 2601–2688. MR 3190296, DOI 10.2140/gt.2013.17.2601
- András Juhász, Holomorphic discs and sutured manifolds, Algebr. Geom. Topol. 6 (2006), 1429–1457. MR 2253454, DOI 10.2140/agt.2006.6.1429
- A. Juhász and D. P. Thurston. Naturality and mapping class groups in Heegaard Floer homology. 2012, math.GT/0210.4996.
- Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
- P. B. Kronheimer and T. S. Mrowka, Monopoles and contact structures, Invent. Math. 130 (1997), no. 2, 209–255. MR 1474156, DOI 10.1007/s002220050183
- Peter Kronheimer and Tomasz Mrowka, Monopoles and three-manifolds, New Mathematical Monographs, vol. 10, Cambridge University Press, Cambridge, 2007. MR 2388043, DOI 10.1017/CBO9780511543111
- Peter Kronheimer and Tomasz Mrowka, Knots, sutures, and excision, J. Differential Geom. 84 (2010), no. 2, 301–364. MR 2652464
- Dusa McDuff, Singularities and positivity of intersections of $J$-holomorphic curves, Holomorphic curves in symplectic geometry, Progr. Math., vol. 117, Birkhäuser, Basel, 1994, pp. 191–215. With an appendix by Gang Liu. MR 1274930, DOI 10.1007/978-3-0348-8508-9_{7}
- Mario J. Micallef and Brian White, The structure of branch points in minimal surfaces and in pseudoholomorphic curves, Ann. of Math. (2) 141 (1995), no. 1, 35–85. MR 1314031, DOI 10.2307/2118627
- John W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes, vol. 44, Princeton University Press, Princeton, NJ, 1996. MR 1367507
- Clifford Henry Taubes, A compendium of pseudoholomorphic beasts in $\Bbb R\times (S^1\times S^2)$, Geom. Topol. 6 (2002), 657–814. MR 1943381, DOI 10.2140/gt.2002.6.657
- Clifford Henry Taubes, Asymptotic spectral flow for Dirac operators, Comm. Anal. Geom. 15 (2007), no. 3, 569–587. MR 2379805
- Clifford Henry Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007), 2117–2202. MR 2350473, DOI 10.2140/gt.2007.11.2117
- Clifford Henry Taubes, The Seiberg-Witten equations and the Weinstein conjecture. II. More closed integral curves of the Reeb vector field, Geom. Topol. 13 (2009), no. 3, 1337–1417. MR 2496048, DOI 10.2140/gt.2009.13.1337
- Clifford Henry Taubes, Embedded contact homology and Seiberg-Witten Floer cohomology I, Geom. Topol. 14 (2010), no. 5, 2497–2581. MR 2746723, DOI 10.2140/gt.2010.14.2497
- Clifford Henry Taubes, Embedded contact homology and Seiberg-Witten Floer cohomology II, Geom. Topol. 14 (2010), no. 5, 2583–2720. MR 2746724, DOI 10.2140/gt.2010.14.2583
- Clifford Henry Taubes, Embedded contact homology and Seiberg-Witten Floer cohomology III, Geom. Topol. 14 (2010), no. 5, 2721–2817. MR 2746725, DOI 10.2140/gt.2010.14.2721
- Clifford Henry Taubes, Embedded contact homology and Seiberg-Witten Floer cohomology IV, Geom. Topol. 14 (2010), no. 5, 2819–2960. MR 2746726, DOI 10.2140/gt.2010.14.2819
- Clifford Henry Taubes, Embedded contact homology and Seiberg-Witten Floer cohomology V, Geom. Topol. 14 (2010), no. 5, 2961–3000. MR 2746727, DOI 10.2140/gt.2010.14.2961
- W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms, Proc. Amer. Math. Soc. 52 (1975), 345–347. MR 375366, DOI 10.1090/S0002-9939-1975-0375366-7
- Karen K. Uhlenbeck, Connections with $L^{p}$ bounds on curvature, Comm. Math. Phys. 83 (1982), no. 1, 31–42. MR 648356
- Alan Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991), no. 2, 241–251. MR 1114405, DOI 10.14492/hokmj/1381413841
- Chris Wendl, A hierarchy of local symplectic filling obstructions for contact 3-manifolds, Duke Math. J. 162 (2013), no. 12, 2197–2283. MR 3102479, DOI 10.1215/00127094-2348333
- Mei-Lin Yau, Vanishing of the contact homology of overtwisted contact 3-manifolds, Bull. Inst. Math. Acad. Sin. (N.S.) 1 (2006), no. 2, 211–229. With an appendix by Yakov Eliashberg. MR 2230587