Constructing symplectic forms on 4–manifolds which vanish on circles

Gay, David T¹ ; Kirby, Robion²

¹ CIRGET, Université du Québec à Montréal, Case Postale 8888, Succursale centre-ville, Montréal, Quebec H3C 3P8, Canada
² Department of Mathematics, University of California, Berkeley, California 94720, USA

Geometry & topology, Tome 8 (2004) no. 2, pp. 743-777.

Résumé

Given a smooth, closed, oriented 4–manifold $X$ and $\alpha \in H_2\left(X,\mathbb{Z}\right)$ such that $\alpha \cdot \alpha >0$, a closed $2$–form $$ is constructed, Poincaré dual to $\alpha$, which is symplectic on the complement of a finite set of unknotted circles $Z$. The number of circles, counted with sign, is given by $d=\left(c_1{\left(s\right)}^2-3\sigma \left(X\right)-2\chi \left(X\right)\right)∕4$, where $s$ is a certain ${spin}^{ℂ}$ structure naturally associated to $\omega$.

Détail

DOI : 10.2140/gt.2004.8.743

Keywords: symplectic, $4$–manifold, $\mathrm{spin}^C$, almost complex, harmonic

Bibliographie

[1] C Bohr, Embedded surfaces and almost complex structures, Proc. Amer. Math. Soc. 128 (2000) 2147

[2] F Ding, H Geiges, A Legendrian surgery presentation of contact 3–manifolds, Math. Proc. Cambridge Philos. Soc. 136 (2004) 583

[3] F Ding, H Geiges, A I Stipsicz, Surgery diagrams for contact 3–manifolds, Turkish J. Math. 28 (2004) 41

[4] Y Eliashberg, Classification of overtwisted contact structures on 3–manifolds, Invent. Math. 98 (1989) 623

[5] Y Eliashberg, Topological characterization of Stein manifolds of dimension gt;2, Internat. J. Math. 1 (1990) 29

[6] D T Gay, Explicit concave fillings of contact three-manifolds, Math. Proc. Cambridge Philos. Soc. 133 (2002) 431

[7] D T Gay, Open books and configurations of symplectic surfaces, Algebr. Geom. Topol. 3 (2003) 569

[8] N Goodman, Contact structures and open books, PhD thesis, University of Texas, Austin (2003)

[9] R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. (2) 148 (1998) 619

[10] R E Gompf, Spinc–structures and homotopy equivalences, Geom. Topol. 1 (1997) 41

[11] F Hirzebruch, H Hopf, Felder von Flächenelementen in 4–dimensionalen Mannigfaltigkeiten, Math. Ann. 136 (1958) 156

[12] K Honda, An openness theorem for harmonic 2–forms on 4–manifolds, Illinois J. Math. 44 (2000) 479

[13] K Honda, Local properties of self-dual harmonic 2–forms on a 4–manifold, J. Reine Angew. Math. 577 (2004) 105

[14] K Honda, On the classification of tight contact structures I, Geom. Topol. 4 (2000) 309

[15] G Mikhalkin, J–holomorphic curves in almost complex surfaces do not always minimize the genus, Proc. Amer. Math. Soc. 125 (1997) 1831

[16] R Scott, Closed self-dual two-forms on four-dimensional handlebodies, PhD thesis, Harvard University (2003)

[17] C H Taubes, The structure of pseudo-holomorphic subvarieties for a degenerate almost complex structure and symplectic form on S1 × B3, Geom. Topol. 2 (1998) 221

[18] A Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991) 241