Abstract
This survey on the topology of Stein manifolds is an extract from the book of Cieliebak and Eliashberg (From Stein to Weinstein and Back—Symplectic Geometry of Affine Complex Manifolds, Colloquium Publications, vol. 59, 2012). It is compiled from two short lecture series given by the first author in 2012 at the Institute for Advanced Study, Princeton, and the Alfréd Rényi Institute of Mathematics, Budapest.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
“Problems” in this survey are meant to be reasonably hard exercises for the reader.
- 2.
“Proofs” in this survey are only sketches of proofs; for details see [7].
- 3.
This figure, and all further figures of this Chapter have been taken from our book [7] with the permission of the American Mathematical Society.
References
M. Abouzaid, P. Seidel, Altering symplectic manifolds by homologous recombination. arXiv:1007.3281
E. Bishop, Mappings of partially analytic spaces. Am. J. Math. 83, 209–242 (1961)
F. Bourgeois, T. Ekholm, Y. Eliashberg, Effect of Legendrian surgery. arXiv:0911.0026
J. Cerf, La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie. Publ. Math. IHÉS 39, 5–173 (1970)
Y. Chekanov, Differential algebra of Legendrian links. Invent. Math. 150(3), 441–483 (2002)
K. Cieliebak, Handle attaching in symplectic homology and the chord conjecture. J. Eur. Math. Soc. 4(2), 115–142 (2002)
K. Cieliebak, Y. Eliashberg, From Stein to Weinstein and Back—Symplectic Geometry of Affine Complex Manifolds. Colloquium Publications, vol. 59 (Amer. Math. Soc., Providence, 2012)
K. Dymara, Legendrian knots in overtwisted contact structures on S 3. Ann. Glob. Anal. Geom. 19(3), 293–305 (2001)
T. Ekholm, J. Etnyre, M. Sullivan, Non-isotopic Legendrian submanifolds in \(\mathbb{R}^{2n+1}\). J. Differ. Geom. 71(1), 85–128 (2005)
Y. Eliashberg, Topological characterization of Stein manifolds of dimension >2. Int. J. Math. 1(1), 29–46 (1990)
Y. Eliashberg, N. Mishachev, Introduction to the h-Principle (Amer. Math. Soc., Providence, 2002)
D. Fuchs, S. Tabachnikov, Invariants of Legendrian and transverse knots in the standard contact space. Topology 36(5), 1025–1053 (1997)
H. Grauert, On Levi’s problem and the imbedding of real-analytic manifolds. Ann. Math. 68, 460–472 (1958)
M. Gromov, Convex integration of differential relations. I. Izv. Akad. Nauk SSSR, Ser. Mat. 37, 329–343 (1973)
M. Gromov, Partial Differential Relations. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 9 (Springer, Berlin, 1986)
A. Hatcher, J. Wagoner, Pseudo-isotopies of Compact Manifolds. Astérisque, vol. 6 (Soc. Math. de France, Paris, 1973)
K. Igusa, The stability theorem for smooth pseudoisotopies. K-Theory 2(1–2) (1988)
P. Landweber, Complex structures on open manifolds. Topology 13, 69–75 (1974)
P. Lisca, G. Matič, Tight contact structures and Seiberg-Witten invariants. Invent. Math. 129, 509–525 (1997)
M. McLean, Lefschetz fibrations and symplectic homology. Geom. Topol. 13(4), 1877–1944 (2009)
J. Milnor, Lectures on the h-Cobordism Theorem (Princeton Univ. Press, Princeton, 1965). Notes by L. Siebenmann and J. Sondow
E. Murphy, Loose Legendrian embeddings in high dimensional contact manifolds. arXiv:1201.2245
R. Narasimhan, Imbedding of holomorphically complete complex spaces. Am. J. Math. 82, 917–934 (1960)
A. Newlander, L. Nirenberg, Complex analytic coordinates in almost complex manifolds. Ann. Math. 65, 391–404 (1957)
R. Richberg, Stetige streng pseudokonvexe Funktionen. Math. Ann. 175, 251–286 (1968)
S. Smale, On the structure of manifolds. Am. J. Math. 84, 387–399 (1962)
H. Whitney, The self-intersections of a smooth n-manifold in 2n-space. Ann. Math. 45, 220–246 (1944)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Copyright jointly owned by the János Bolyai Mathematical Society and Springer
About this chapter
Cite this chapter
Cieliebak, K., Eliashberg, Y. (2014). Stein Structures: Existence and Flexibility. In: Bourgeois, F., Colin, V., Stipsicz, A. (eds) Contact and Symplectic Topology. Bolyai Society Mathematical Studies, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-02036-5_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-02036-5_8
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-02035-8
Online ISBN: 978-3-319-02036-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)