No Access

A topological obstruction to the removal of a degenerate complex tangent and some related homotopy and homology groups

Ali M. Elgindi

Department of Mathematics, Alfaisal University, Riyadh, Saudi Arabia

https://doi.org/10.1142/S0129167X15500251Cited by:3 (Source: Crossref)

Abstract

In this paper, we derive a topological obstruction to the removal of an isolated degenerate complex tangent to an embedding of a 3-manifold into ℂ³ (without affecting the structure of the remaining complex tangents). We demonstrate how the vanishing of this obstruction is both a necessary and sufficient condition for the (local) removal of the isolated complex tangent. The obstruction is a certain homotopy class of the space 𝕐 consisting of totally real 3-planes in the Grassmannian of real 3-planes in ℂ³(= ℝ⁶). We further compute additional homotopy and homology groups for the space 𝕐 and of its complement 𝕎 consisting of "partially complex" 3-planes in ℂ³.

Keywords:

AMSC: 32C09, 32V40, 32W99

References

P. Ahern and W. Rudin, Proc. Amer. Math. Soc. 94(5), 460 (1985). Web of Science, Google Scholar
R. Bott , Lectures on K(X) ( W. A. Benjamin , 1969 ) . Google Scholar
K. Cieliebak and Y. Eliashberg , From Stein to Weinstein and Back: Symplectic Geometry of Affine Complex Manifolds , AMS Colloquium Publications 59 ( American Mathematical Society , Providence, RI , 2013 ) . Google Scholar
A. V. Domrin , Izv. Ross. Akad. Nauk Ser. Mat. (English transl.) (5) 59 , 899 ( 1995 ) . Google Scholar
A. Elgindi , New York J. Math. 18 , 295 ( 2012 ) . Web of Science, Google Scholar
F. Forstneric , Expo. Math. 4 , 243 ( 1986 ) . Google Scholar
M. Gromov , Partial Differential Relations ( Springer-Verlag , 1986 ) . Crossref, Google Scholar
A. Hatcher , Algebraic Topology ( Cambridge University Press , 2002 ) . Google Scholar
A. Hatcher, Vector bundles and k-theory, http://www.math.cornell.edu/hatcher . Google Scholar
H. F. Lai , Trans. Amer. Math. Soc. 172 , 1 ( 1972 ) . Crossref, Web of Science, Google Scholar
J. Lee , Introduction to Smooth Manifolds , Springer Graduate Texts in Mathematics 218 ( Springer , 2012 ) . Crossref, Google Scholar
J. Milnor and J. Stasheff , Characteristic Classes ( Princeton University Press , 1974 ) . Crossref, Google Scholar
R. Schultz , Manuscripta Math. 24 , 229 ( 1978 ) . Crossref, Web of Science, Google Scholar
M. Slapar, Bull. Austral. Math. Soc. 88(1), 64 (2013). Crossref, Web of Science, Google Scholar
S. Webster , Comment. Math. Helv. 60 , 193 ( 1985 ) . Crossref, Web of Science, Google Scholar