×

Taming the pseudoholomorphic beasts in \(\mathbb{R} \times (S^1 \times S^2)\). (English) Zbl 1452.53074

Summary: For a closed oriented smooth \(4\)-manifold \(X\) with \(b^2_+(X)>0\), the Seiberg-Witten invariants are well-defined. Taubes’ “\( \operatorname{SW}=\operatorname{Gr} \)” theorem asserts that if \(X\) carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes’ Gromov invariants. In the absence of a symplectic form, there are still nontrivial closed self-dual \(2\)-forms which vanish along a disjoint union of circles and are symplectic elsewhere. This paper and its sequel describe well-defined integral counts of pseudoholomorphic curves in the complement of the zero set of such near-symplectic \(2\)-forms, and it is shown that they recover the Seiberg-Witten invariants over \(\mathbb{Z}/2\mathbb{Z} \). This is an extension of “\( \operatorname{SW}=\operatorname{Gr} \)” to nonsymplectic \(4\)-manifolds.
The main result of this paper asserts the following. Given a suitable near-symplectic form \(\omega\) and tubular neighborhood \(\mathcal{N}\) of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the symplectic cobordism \((X-\mathcal{N},\omega)\) which are asymptotic to certain Reeb orbits on the ends. They can be packaged together to form “near-symplectic” Gromov invariants as a function of spin-c structures on \(X\).

MSC:

53D42 Symplectic field theory; contact homology
57R57 Applications of global analysis to structures on manifolds
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 10.2140/gt.2005.9.1043 · Zbl 1077.53069 · doi:10.2140/gt.2005.9.1043
[2] ; Bourgeois, New perspectives and challenges in symplectic field theory. CRM Proc. Lecture Notes, 49, 45 (2009)
[3] 10.2140/gt.2003.7.799 · Zbl 1131.53312 · doi:10.2140/gt.2003.7.799
[4] 10.1007/s00209-004-0656-x · Zbl 1060.53080 · doi:10.1007/s00209-004-0656-x
[5] 10.2140/agt.2013.13.2239 · Zbl 1279.53080 · doi:10.2140/agt.2013.13.2239
[6] 10.2140/gt.2004.8.743 · Zbl 1054.57027 · doi:10.2140/gt.2004.8.743
[7] 10.2307/121005 · Zbl 0919.57012 · doi:10.2307/121005
[8] 10.1007/BF01388806 · Zbl 0592.53025 · doi:10.1007/BF01388806
[9] 10.1515/crll.2004.2004.577.105 · Zbl 1065.53066 · doi:10.1515/crll.2004.2004.577.105
[10] 10.1216/rmjm/1181069872 · Zbl 1071.58002 · doi:10.1216/rmjm/1181069872
[11] ; Hutchings, New perspectives and challenges in symplectic field theory. CRM Proc. Lecture Notes, 49, 263 (2009)
[12] 10.1007/978-3-319-02036-5_9 · Zbl 1432.53126 · doi:10.1007/978-3-319-02036-5_9
[13] 10.2140/gt.2016.20.1085 · Zbl 1338.53119 · doi:10.2140/gt.2016.20.1085
[14] 10.2140/gt.2006.10.169 · Zbl 1101.53053 · doi:10.2140/gt.2006.10.169
[15] 10.4310/JSG.2007.v5.n1.a5 · Zbl 1157.53047 · doi:10.4310/JSG.2007.v5.n1.a5
[16] 10.4310/JSG.2009.v7.n1.a2 · Zbl 1193.53183 · doi:10.4310/JSG.2009.v7.n1.a2
[17] 10.2140/gt.2009.13.901 · Zbl 1169.53065 · doi:10.2140/gt.2009.13.901
[18] 10.1017/CBO9780511543111 · doi:10.1017/CBO9780511543111
[19] 10.4310/CAG.1997.v5.n3.a6 · Zbl 0901.53028 · doi:10.4310/CAG.1997.v5.n3.a6
[20] 10.2307/1990934 · Zbl 0723.53019 · doi:10.2307/1990934
[21] 10.1007/978-94-017-1667-3_6 · doi:10.1007/978-94-017-1667-3_6
[22] 10.4310/JSG.2006.v4.n3.a1 · Zbl 1132.53314 · doi:10.4310/JSG.2006.v4.n3.a1
[23] 10.1142/9789814350112_0010 · doi:10.1142/9789814350112_0010
[24] 10.4310/MRL.1995.v2.n2.a10 · Zbl 0854.57020 · doi:10.4310/MRL.1995.v2.n2.a10
[25] 10.4310/jdg/1214459411 · Zbl 0883.57020 · doi:10.4310/jdg/1214459411
[26] 10.1090/S0894-0347-96-00211-1 · Zbl 0867.53025 · doi:10.1090/S0894-0347-96-00211-1
[27] ; Taubes, Proceedings of the International Congress of Mathematicians, II, 493 (1998)
[28] 10.4310/SDG.1996.v3.n1.a5 · doi:10.4310/SDG.1996.v3.n1.a5
[29] 10.2140/gt.1998.2.221 · Zbl 0908.53013 · doi:10.2140/gt.1998.2.221
[30] 10.4310/AJM.1999.v3.n1.a11 · Zbl 0972.53055 · doi:10.4310/AJM.1999.v3.n1.a11
[31] 10.2140/gt.1999.3.167 · Zbl 1027.53111 · doi:10.2140/gt.1999.3.167
[32] ; Taubes, Seiberg-Witten and Gromov invariants for symplectic 4-manifolds. First Int. Press Lect. Ser., 2 (2000) · Zbl 0967.57001
[33] 10.4310/SDG.2002.v7.n1.a19 · doi:10.4310/SDG.2002.v7.n1.a19
[34] 10.2140/gt.2002.6.657 · Zbl 1021.32008 · doi:10.2140/gt.2002.6.657
[35] 10.4310/MRL.2006.v13.n4.a6 · Zbl 1151.57032 · doi:10.4310/MRL.2006.v13.n4.a6
[36] 10.2140/gt.2006.10.785 · Zbl 1134.53045 · doi:10.2140/gt.2006.10.785
[37] 10.2140/gt.2006.10.1855 · Zbl 1161.53075 · doi:10.2140/gt.2006.10.1855
[38] 10.2140/gt.2010.14.2497 · Zbl 1275.57037 · doi:10.2140/gt.2010.14.2497
[39] 10.2140/gt.2010.14.2721 · Zbl 1276.57025 · doi:10.2140/gt.2010.14.2721
[40] 10.2140/gt.2010.14.2961 · Zbl 1276.57027 · doi:10.2140/gt.2010.14.2961
[41] 10.4171/CMH/199 · Zbl 1207.32021 · doi:10.4171/CMH/199
[42] 10.4310/MRL.1994.v1.n6.a13 · Zbl 0867.57029 · doi:10.4310/MRL.1994.v1.n6.a13
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.