CR structures on open manifolds

Jacobowitz, Howard; Landweber, Peter

doi:10.1090/proc/12700

CR structures on open manifolds
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by Howard Jacobowitz and Peter Landweber PDF

Proc. Amer. Math. Soc. 144 (2016), 235-248 Request permission

Abstract:

We show that the vanishing of the higher dimensional homology groups of a manifold ensures that every almost CR structure of codimension $k$ may be homotoped to a CR structure. This result is proved by adapting a method Haefliger used to study foliations (and previously applied to study the relation between almost complex and complex structures on manifolds) to the case of (almost) CR structures on open manifolds.

References

Masahisa Adachi, Embeddings and immersions, Translations of Mathematical Monographs, vol. 124, American Mathematical Society, Providence, RI, 1993. Translated from the 1984 Japanese original by Kiki Hudson. MR 1225100, DOI 10.1090/mmono/124
M. Salah Baouendi, Peter Ebenfelt, and Linda Preiss Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series, vol. 47, Princeton University Press, Princeton, NJ, 1999. MR 1668103, DOI 10.1515/9781400883967
Raoul Bott, On a topological obstruction to integrability, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 127–131. MR 0266248
Raoul Bott, On topological obstructions to integrability, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 27–36. MR 0425983
Raoul Bott, Lectures on characteristic classes and foliations, Lectures on algebraic and differential topology (Second Latin American School in Math., Mexico City, 1971) Lecture Notes in Math., Vol. 279, Springer, Berlin, 1972, pp. 1–94. Notes by Lawrence Conlon, with two appendices by J. Stasheff. MR 0362335
Beno Eckmann and Alfred Frölicher, Sur l’intégrabilité des structures presque complexes, C. R. Acad. Sci. Paris 232 (1951), 2284–2286 (French). MR 42202
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
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 and Marko Slapar, Stein structures and holomorphic mappings, Math. Z. 256 (2007), no. 3, 615–646. MR 2299574, DOI 10.1007/s00209-006-0093-0
M. L. Gromov, Convex integration of differential relations. I, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 329–343 (Russian). MR 0413206
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
A. Haefliger, Feuilletages sur les variétés ouvertes, Topology 9 (1970), 183–194 (French). MR 263104, DOI 10.1016/0040-9383(70)90040-6
André Haefliger, Homotopy and integrability, Manifolds–Amsterdam 1970 (Proc. Nuffic Summer School), Lecture Notes in Mathematics, Vol. 197, Springer, Berlin, 1971, pp. 133–163. MR 0285027
H. Jacobowitz, Convex integration and the h-principle. Lecture Notes Series, 55. Seoul National University, Seoul, 2011.
Howard Jacobowitz and Peter Landweber, Manifolds admitting generic immersions into $\Bbb C^N$, Asian J. Math. 11 (2007), no. 1, 151–165. MR 2304588, DOI 10.4310/AJM.2007.v11.n1.a14
Peter S. Landweber, Complex structures on open manifolds, Topology 13 (1974), 69–75. MR 339210, DOI 10.1016/0040-9383(74)90039-1
H. Blaine Lawson Jr., Foliations, Bull. Amer. Math. Soc. 80 (1974), 369–418. MR 343289, DOI 10.1090/S0002-9904-1974-13432-4
H. Blaine Lawson Jr., The quantitative theory of foliations, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 27, American Mathematical Society, Providence, R.I., 1977. Expository lectures from the CBMS Regional Conference held at Washington University, St. Louis, Mo., January 6–10, 1975. MR 0448368
Paulette Libermann, Problèmes d’équivalence relatifs à une structure presque complexe sur une variété à quatre dimensions, Acad. Roy. Belgique. Bull. Cl. Sci. (5) 36 (1950), 742–755 (French). MR 0040791
John Milnor, Collected papers of John Milnor. IV, American Mathematical Society, Providence, RI, 2009. Homotopy, homology and manifolds; Edited by John McCleary. MR 2590677
A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. of Math. (2) 65 (1957), 391–404. MR 88770, DOI 10.2307/1970051
Louis Nirenberg, Lectures on linear partial differential equations, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 17, American Mathematical Society, Providence, R.I., 1973. Expository Lectures from the CBMS Regional Conference held at the Texas Technological University, Lubbock, Tex., May 22–26, 1972. MR 0450755
Anthony Phillips, Submersions of open manifolds, Topology 6 (1967), 171–206. MR 208611, DOI 10.1016/0040-9383(67)90034-1
N. E. Steenrod, Homology with local coefficients, Ann. of Math. (2) 44 (1943), 610–627. MR 9114, DOI 10.2307/1969099
Norman Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, vol. 14, Princeton University Press, Princeton, N. J., 1951. MR 0039258
George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508