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

This article is cited in 10 scientific papers (total in 10 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

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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

Citation in format AMSBIB
\by V.~V.~Shevchishin
\paper Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groups
\jour Izv. RAN. Ser. Mat.
\yr 2009
\vol 73
\issue 4
\pages 153--224
\jour Izv. Math.
\yr 2009
\vol 73
\issue 4
\pages 797--859

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  • https://doi.org/10.4213/im2638
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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
    9. Anderson J.T., Gupta P., Stout E.L., “the Rational Hull of Rudin'S Klein Bottle”, Proc. Amer. Math. Soc., 147:9 (2019), 3859–3866  crossref  isi
    10. Shevchishin V. Smirnov G., “Symplectic Triangle Inequality”, Proc. Amer. Math. Soc., 148:4 (2020), 1389–1397  crossref  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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