Totally real immersions of surfaces
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- by Andrzej Derdzinski and Tadeusz Januszkiewicz PDF
- Trans. Amer. Math. Soc. 362 (2010), 53-115 Request permission
Abstract:
Totally real immersions $f$ of a closed real surface $\varSigma$ in an almost complex surface $M$ are completely classified, up to homotopy through totally real immersions, by suitably defined homotopy classes $\mathfrak {M}(f)$ of mappings from $\varSigma$ into a specific real $5$-manifold $E(M)$, while $\mathfrak {M}(f)$ themselves are subject to a single cohomology constraint. This follows from Gromov’s observation that totally real immersions satisfy the $h$-principle. For the receiving complex surfaces $\mathbf {C}^2$, $\mathbf {C} \mathrm {P} ^1\times \mathbf {C} \mathrm {P}^1$, $\mathbf {C} \mathrm {P}^2$ and $\mathbf {C} \mathrm {P}^2 \# m \overline {\mathbf {C} \mathrm {P}^2}$, $m=1$, $2$, …, $7$, and all $\varSigma$ (or, $\mathbf {C} \mathrm {P}^2\# 8\overline {\mathbf {C} \mathrm {P}}$ and all orientable $\varSigma$), we illustrate the above nonconstructive result with explicit examples of immersions realizing all possible equivalence classes. We also determine which equivalence classes contain totally real embeddings, and provide examples of such embeddings for all classes that contain them.References
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Additional Information
- Andrzej Derdzinski
- Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
- Email: andrzej@math.ohio-state.edu
- Tadeusz Januszkiewicz
- Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210 – and – Mathematical Institute, Wrocław University, 50-384 Wrocław, Poland – and – Mathematical Institute, Polish Academy of Sciences, 51-617 Wrocław, Poland
- Email: tjan@math.ohio-state.edu
- Received by editor(s): January 9, 2007
- Published electronically: August 17, 2009
- Additional Notes: The second author was partially supported by NSF grant no. DMS-0405825
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 53-115
- MSC (2000): Primary 53C15, 53C42; Secondary 32Q60
- DOI: https://doi.org/10.1090/S0002-9947-09-04940-X
- MathSciNet review: 2550145