Book Review
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3362820
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Book Information:
Authors:
Kai Cieliebak and
Yakov Eliashberg
Title:
From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds
Additional book information:
AMS Colloquium Publications, vol.~59,
American Mathematical Society,
Providence, RI,
2012,
xii+364 pp.,
ISBN 978-0-8218-8533-8,
US $78.,
hardcover
M. Abouzaid and P. Seidel, Altering symplectic manifolds by homologous recombination, 2010. arXiv:1007.3281
Frédéric Bourgeois, Tobias Ekholm, and Yasha Eliashberg, Effect of Legendrian surgery, Geom. Topol. 16 (2012), no. 1, 301–389. With an appendix by Sheel Ganatra and Maksim Maydanskiy. MR 2916289, DOI 10.2140/gt.2012.16.301
Henri Cartan, Variétés analytiques complexes et cohomologie, Colloque sur les fonctions de plusieurs variables, tenu à Bruxelles, 1953, Georges Thone, Liège; Masson & Cie, Paris, 1953, pp. 41–55 (French). MR 0064154
K. Cieliebak, Subcritical Stein manifolds are split, 2002. arXiv:math/0204351
Kai Cieliebak and Yakov Eliashberg, From Stein to Weinstein and back, American Mathematical Society Colloquium Publications, vol. 59, American Mathematical Society, Providence, RI, 2012. Symplectic geometry of affine complex manifolds. MR 3012475, DOI 10.1090/coll/059
K. Cieliebak and Y. Eliashberg, Flexible Weinstein manifolds, 2013. arXiv:1305.1635
Kai Cieliebak and Yakov Eliashberg, Stein structures: existence and flexibility, Contact and symplectic topology, Bolyai Soc. Math. Stud., vol. 26, János Bolyai Math. Soc., Budapest, 2014, pp. 357–388. MR 3220946, DOI 10.1007/978-3-319-02036-5_{8}
K. Cieliebak, A. Floer, and H. Hofer, Symplectic homology. II. A general construction, Math. Z. 218 (1995), no. 1, 103–122. MR 1312580, DOI 10.1007/BF02571891
Yakov Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, Internat. J. Math. 1 (1990), no. 1, 29–46. MR 1044658, DOI 10.1142/S0129167X90000034
Yakov Eliashberg, Recent advances in symplectic flexibility, Bull. Amer. Math. Soc. (N.S.) 52 (2015), no. 1, 1–26. MR 3286479, DOI 10.1090/S0273-0979-2014-01470-3
Yakov Eliashberg and Mikhael Gromov, Convex symplectic manifolds, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989) Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 135–162. MR 1128541, DOI 10.1090/pspum/052.2/1128541
Yakov Eliashberg and Mikhael Gromov, Embeddings of Stein manifolds of dimension $n$ into the affine space of dimension $3n/2+1$, Ann. of Math. (2) 136 (1992), no. 1, 123–135. MR 1173927, DOI 10.2307/2946547
Y. Eliashberg and N. Mishachev, Introduction to the $h$-principle, Graduate Studies in Mathematics, vol. 48, American Mathematical Society, Providence, RI, 2002. MR 1909245, DOI 10.1090/gsm/048
Franc Forstnerič, Stein manifolds and holomorphic mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 56, Springer, Heidelberg, 2011. The homotopy principle in complex analysis. MR 2975791, DOI 10.1007/978-3-642-22250-4
Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738, DOI 10.1017/CBO9780511611438
Hans Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460–472. MR 98847, DOI 10.2307/1970257
Hans Grauert and Reinhold Remmert, Theory of Stein spaces, Classics in Mathematics, Springer-Verlag, Berlin, 2004. Translated from the German by Alan Huckleberry; Reprint of the 1979 translation. MR 2029201, DOI 10.1007/978-3-642-18921-0
M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, DOI 10.1007/BF01388806
Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505, DOI 10.1007/978-3-662-02267-2
M. L. Gromov, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 707–734 (Russian). MR 0263103
M. L. Gromov, A topological technique for the construction of solutions of differential equations and inequalities, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 221–225. MR 0420697
M. L. Gromov and Ja. M. Èliašberg, Elimination of singularities of smooth mappings, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 600–626 (Russian). MR 0301748
M. L. Gromov and Ja. M. Èliašberg, Nonsingular mappings of Stein manifolds, Funkcional. Anal. i Priložen. 5 (1971), no. 2, 82–83 (Russian). MR 0301236
Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, AMS Chelsea Publishing, Providence, RI, 2009. Reprint of the 1965 original. MR 2568219, DOI 10.1090/chel/368
Morris W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR 119214, DOI 10.1090/S0002-9947-1959-0119214-4
Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625, DOI 10.1090/chel/340
Finnur Lárusson, Mapping cylinders and the Oka principle, Indiana Univ. Math. J. 54 (2005), no. 4, 1145–1159. MR 2164421, DOI 10.1512/iumj.2005.54.2731
J. Alexander Lees, On the classification of Lagrange immersions, Duke Math. J. 43 (1976), no. 2, 217–224. MR 410764
Maksim Maydanskiy, Exotic symplectic manifolds from Lefschetz fibrations, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–Massachusetts Institute of Technology. MR 2717731
Maksim Maydanskiy and Paul Seidel, Lefschetz fibrations and exotic symplectic structures on cotangent bundles of spheres, J. Topol. 3 (2010), no. 1, 157–180. MR 2608480, DOI 10.1112/jtopol/jtq003
Mark McLean, Lefschetz fibrations and symplectic homology, Geom. Topol. 13 (2009), no. 4, 1877–1944. MR 2497314, DOI 10.2140/gt.2009.13.1877
J. Milnor, Morse theory, Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. Based on lecture notes by M. Spivak and R. Wells. MR 0163331
E. Murphy, Loose Legendrian embeddings in high dimensional contact manifolds, 2012. arXiv:1201.2245
K. Oka, Deuxième problème de Cousin. III, Journ. Sci. Hiroshima Univ., 9:7–19, 1939.
Kiyoshi Oka, Collected papers, Springer-Verlag, Berlin, 1984. Translated from the French by R. Narasimhan; With commentaries by H. Cartan; Edited by R. Remmert. MR 754337, DOI 10.1007/978-3-642-15083-8
Paul Seidel and Ivan Smith, The symplectic topology of Ramanujam’s surface, Comment. Math. Helv. 80 (2005), no. 4, 859–881. MR 2182703, DOI 10.4171/CMH/37
Jean-Pierre Serre, Quelques problèmes globaux relatifs aux variétés de Stein, Colloque sur les fonctions de plusieurs variables, tenu à Bruxelles, 1953, Georges Thone, Liège; Masson & Cie, Paris, 1953, pp. 57–68 (French). MR 0064155
Jean-Pierre Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 1–42 (French). MR 82175
Stephen Smale, A classification of immersions of the two-sphere, Trans. Amer. Math. Soc. 90 (1958), 281–290. MR 104227, DOI 10.1090/S0002-9947-1959-0104227-9
Stephen Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math. (2) 69 (1959), 327–344. MR 105117, DOI 10.2307/1970186
C. Viterbo, Functors and computations in Floer homology with applications. I, Geom. Funct. Anal. 9 (1999), no. 5, 985–1033. MR 1726235, DOI 10.1007/s000390050106
Alan Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991), no. 2, 241–251. MR 1114405, DOI 10.14492/hokmj/1381413841
Hassler Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937), 276–284. MR 1556973
References
- M. Abouzaid and P. Seidel, Altering symplectic manifolds by homologous recombination, 2010. arXiv:1007.3281
- Frédéric Bourgeois, Tobias Ekholm, and Yasha Eliashberg, Effect of Legendrian surgery, Geom. Topol. 16 (2012), no. 1, 301–389. With an appendix by Sheel Ganatra and Maksim Maydanskiy. MR 2916289, DOI 10.2140/gt.2012.16.301
- Henri Cartan, Variétés analytiques complexes et cohomologie, Colloque sur les fonctions de plusieurs variables, tenu à Bruxelles, 1953, Georges Thone, Liège; Masson & Cie, Paris, 1953, pp. 41–55 (French). MR 0064154 (16,235a)
- K. Cieliebak, Subcritical Stein manifolds are split, 2002. arXiv:math/0204351
- K. Cieliebak and Y. Eliashberg, From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds, AMS Colloquium Publications, vol. 59, American Mathematical Society, Providence, RI, 2012. MR 3012475
- K. Cieliebak and Y. Eliashberg, Flexible Weinstein manifolds, 2013. arXiv:1305.1635
- Kai Cieliebak and Yakov Eliashberg, Stein structures: existence and flexibility, Contact and symplectic topology, Bolyai Soc. Math. Stud., vol. 26, János Bolyai Math. Soc., Budapest, 2014, pp. 357–388. MR 3220946, DOI 10.1007/978-3-319-02036-5_8
- K. Cieliebak, A. Floer, and H. Hofer, Symplectic homology. II. A general construction, Math. Z. 218 (1995), no. 1, 103–122. MR 1312580 (95m:58055), DOI 10.1007/BF02571891
- Yakov Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, Internat. J. Math. 1 (1990), no. 1, 29–46. MR 1044658 (91k:32012), DOI 10.1142/S0129167X90000034
- Yakov Eliashberg, Recent advances in symplectic flexibility, Bull. Amer. Math. Soc. (N.S.) 52 (2015), no. 1, 1–26. MR 3286479, DOI 10.1090/S0273-0979-2014-01470-3
- Yakov Eliashberg and Mikhael Gromov, Convex symplectic manifolds, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), Proc. Sympos. Pure Math., vol. 52, Amer. Math. Soc., Providence, RI, 1991, pp. 135–162. MR 1128541 (93f:58073), DOI 10.1090/pspum/052.2/1128541
- Yakov Eliashberg and Mikhael Gromov, Embeddings of Stein manifolds of dimension $n$ into the affine space of dimension $3n/2+1$, Ann. of Math. (2) 136 (1992), no. 1, 123–135. MR 1173927 (93g:32037), DOI 10.2307/2946547
- Y. Eliashberg and N. Mishachev, Introduction to the $h$-principle, Graduate Studies in Mathematics, vol. 48, American Mathematical Society, Providence, RI, 2002. MR 1909245 (2003g:53164)
- Franc Forstnerič, Stein manifolds and holomorphic mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 56, Springer, Heidelberg, 2011. The homotopy principle in complex analysis. MR 2975791, DOI 10.1007/978-3-642-22250-4
- Hansjörg Geiges, An introduction to contact topology, Cambridge Studies in Advanced Mathematics, vol. 109, Cambridge University Press, Cambridge, 2008. MR 2397738 (2008m:57064), DOI 10.1017/CBO9780511611438
- Hans Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460–472. MR 0098847 (20 \#5299)
- Hans Grauert and Reinhold Remmert, Theory of Stein spaces, Classics in Mathematics, Springer-Verlag, Berlin, 2004. Translated from the German by Alan Huckleberry; Reprint of the 1979 translation. MR 2029201 (2004k:32015), DOI 10.1007/978-3-642-18921-0
- M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718 (87j:53053), DOI 10.1007/BF01388806
- Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505 (90a:58201), DOI 10.1007/978-3-662-02267-2
- M. L. Gromov, Stable mappings of foliations into manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 707–734 (Russian). MR 0263103 (41 \#7708)
- M. L. Gromov, A topological technique for the construction of solutions of differential equations and inequalities, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 221–225. MR 0420697 (54 \#8709)
- M. L. Gromov and Ja. M. Èliašberg, Elimination of singularities of smooth mappings, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 600–626 (Russian). MR 0301748 (46 \#903)
- M. L. Gromov and Ja. M. Èliašberg, Nonsingular mappings of Stein manifolds, Funkcional. Anal. i Priložen. 5 (1971), no. 2, 82–83 (Russian). MR 0301236 (46 \#394)
- Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, AMS Chelsea Publishing, Providence, RI, 2009. Reprint of the 1965 original. MR 2568219 (2010j:32001)
- Morris W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR 0119214 (22 \#9980)
- Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625 (2002e:32001)
- Finnur Lárusson, Mapping cylinders and the Oka principle, Indiana Univ. Math. J. 54 (2005), no. 4, 1145–1159. MR 2164421 (2006f:32035), DOI 10.1512/iumj.2005.54.2731
- J. Alexander Lees, On the classification of Lagrange immersions, Duke Math. J. 43 (1976), no. 2, 217–224. MR 0410764 (53 \#14509)
- Maksim Maydanskiy, Exotic symplectic manifolds from Lefschetz fibrations, ProQuest LLC, Ann Arbor, MI, 2009. Thesis (Ph.D.)–Massachusetts Institute of Technology. MR 2717731
- Maksim Maydanskiy and Paul Seidel, Lefschetz fibrations and exotic symplectic structures on cotangent bundles of spheres, J. Topol. 3 (2010), no. 1, 157–180. MR 2608480 (2011j:53176), DOI 10.1112/jtopol/jtq003
- Mark McLean, Lefschetz fibrations and symplectic homology, Geom. Topol. 13 (2009), no. 4, 1877–1944. MR 2497314 (2011d:53224), DOI 10.2140/gt.2009.13.1877
- J. Milnor, Morse theory, Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, N.J., 1963. MR 0163331 (29 \#634)
- E. Murphy, Loose Legendrian embeddings in high dimensional contact manifolds, 2012. arXiv:1201.2245
- K. Oka, Deuxième problème de Cousin. III, Journ. Sci. Hiroshima Univ., 9:7–19, 1939.
- Kiyoshi Oka, Collected papers, Springer-Verlag, Berlin, 1984. Translated from the French by R. Narasimhan; With commentaries by H. Cartan; Edited by R. Remmert. MR 754337 (85i:01037), DOI 10.1007/978-3-642-15083-8
- Paul Seidel and Ivan Smith, The symplectic topology of Ramanujam’s surface, Comment. Math. Helv. 80 (2005), no. 4, 859–881. MR 2182703 (2006j:53127), DOI 10.4171/CMH/37
- Jean-Pierre Serre, Quelques problèmes globaux relatifs aux variétés de Stein, Colloque sur les fonctions de plusieurs variables, tenu à Bruxelles, 1953, Georges Thone, Liège; Masson & Cie, Paris, 1953, pp. 57–68 (French). MR 0064155 (16,235b)
- Jean-Pierre Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble 6 (1955–1956), 1–42 (French). MR 0082175 (18,511a)
- Stephen Smale, A classification of immersions of the two-sphere, Trans. Amer. Math. Soc. 90 (1958), 281–290. MR 0104227 (21 \#2984)
- Stephen Smale, The classification of immersions of spheres in Euclidean spaces, Ann. of Math. (2) 69 (1959), 327–344. MR 0105117 (21 \#3862)
- C. Viterbo, Functors and computations in Floer homology with applications. I, Geom. Funct. Anal. 9 (1999), no. 5, 985–1033. MR 1726235 (2000j:53115), DOI 10.1007/s000390050106
- Alan Weinstein, Contact surgery and symplectic handlebodies, Hokkaido Math. J. 20 (1991), no. 2, 241–251. MR 1114405 (92g:53028), DOI 10.14492/hokmj/1381413841
- Hassler Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937), 276–284. MR 1556973
Review Information:
Reviewer:
Alexandru Oancea
Affiliation:
Université Pierre et Marie Curie, Institut de Mathématiques de Jussieu-Paris Rive Gauche
Email:
alexandru.oancea@imj-prg.fr
Journal:
Bull. Amer. Math. Soc.
52 (2015), 521-530
DOI:
https://doi.org/10.1090/S0273-0979-2015-01487-4
Published electronically:
February 19, 2015
Review copyright:
© Copyright 2015
American Mathematical Society
