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Izv. RAN. Ser. Mat., 2009, Volume 73, Issue 4, Pages 153–224 (Mi izv2638)  

This article is cited in 8 scientific papers (total in 8 papers)

Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groups

V. V. Shevchishin

University of Bonn, Mathematical Institute

Abstract: In this paper we prove the non-existence of Lagrangian embeddings of the Klein bottle $K$ in $\mathbb{R}^4$ and $\mathbb{C}\mathbb{P}^2$. We exploit the existence of a special embedding of $K$ in a symplectic Lefschetz pencil $\operatorname{pr}\colon X \to S^2$ and study its monodromy. As the main technical tool, we develop the combinatorial theory of mapping class groups. The results obtained enable us to show that in the case when the class $[K]\in\mathsf{H}_2(X,\mathbb{Z}_2)$ is trivial, the monodromy of $\operatorname{pr}\colon X\to S^2$ must be of a special form. Finally, we show that such a monodromy cannot be realized in $\mathbb{C}\mathbb{P}^2$.

Keywords: symplectic geometry, Lagrangian submanifold, Lefschetz pencil, monodromy, mapping class group, Coxeter system, Artin–Brieskorn group.

DOI: https://doi.org/10.4213/im2638

Full text: PDF file (1115 kB)
References: PDF file   HTML file

English version:
Izvestiya: Mathematics, 2009, 73:4, 797–859

Bibliographic databases:

UDC: 513.8+515.1
MSC: 57R17, 53D12, 20F36, 20F55
Received: 26.03.2007

Citation: V. V. Shevchishin, “Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groups”, Izv. RAN. Ser. Mat., 73:4 (2009), 153–224; Izv. Math., 73:4 (2009), 797–859

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. Yu. Nemirovski, “Homology class of a Lagrangian Klein bottle”, Izv. Math., 73:4 (2009), 689–698  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. S. Nemirovski, “Lagrangian Klein bottles in $\mathbb{R}^{2n}$”, Geom. Funct. Anal., 19:3 (2009), 902–909  crossref  mathscinet  zmath  isi
    3. A. E. Mironov, T. E. Panov, “Intersections of Quadrics, Moment-Angle Manifolds, and Hamiltonian-Minimal Lagrangian Embeddings”, Funct. Anal. Appl., 47:1 (2013), 38–49  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. T. E. Panov, “Geometric structures on moment-angle manifolds”, Russian Math. Surveys, 68:3 (2013), 503–568  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    5. I. Castro, A. M. Lerma, “The Clifford torus as a self-shrinker for the Lagrangian mean curvature flow”, Int. Math. Res. Notices, 2014, no. 6, 1515–1527  crossref  mathscinet  zmath  isi
    6. Stefan Nemirovski, Kyler Siegel, “Rationally convex domains and singular Lagrangian surfaces in
      $$\mathbb {C}^2$$
      C 2”, Invent. math, 2015  crossref  mathscinet
    7. Damian M., “on the Topology of Monotone Lagrangian Submanifolds”, Ann. Sci. Ec. Norm. Super., 48:1 (2015), 237–252  crossref  mathscinet  zmath  isi
    8. Latschev J., “Fukaya'S Work on Lagrangian Embeddings”, Free Loop Spaces in Geometry and Topology, Irma Lectures in Mathematics and Theoretical Physics, ed. Latschev J. Oancea A., Eur. Math. Soc., 2015, 243–270  mathscinet  zmath  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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