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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Totally real immersions of surfaces


Authors: Andrzej Derdzinski and Tadeusz Januszkiewicz
Journal: Trans. Amer. Math. Soc. 362 (2010), 53-115
MSC (2000): Primary 53C15, 53C42; Secondary 32Q60
Published electronically: August 17, 2009
MathSciNet review: 2550145
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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}$P$ ^1\times \mathbf{C}$P$ ^1$, $ \mathbf{C}$P$ ^2$ and $ \mathbf{C}$P$ ^2 \char93 m \overline{\mathbf{C}\text{\rm P}^2}$, $ m=1,2,\dots,7$, and all $ \varSigma$ (or, $ \mathbf{C}$P$ ^2\char93 8\overline{\mathbf{C}\text{\rm 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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04940-X
PII: S 0002-9947(09)04940-X
Keywords: Totally real immersion, pseudoholomorphic immersion, Maslov invariant, $h$-principle
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
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.