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Izv. Akad. Nauk SSSR Ser. Mat., 1981, Volume 45, Issue 3, Pages 467–490 (Mi izv2377)  

This article is cited in 4 scientific papers (total in 6 papers)

Some homology classes in the space of closed curves in the $n$-dimensional sphere

D. V. Anosov


Abstract: The $(n-1)$-dimensional $\mod2$ cycle generated by the great circles passing through two fixed, diametrically opposite points in the -dimensional sphere $S^n$ is considered in the space $\Pi S^n$ of nonoriented, nonparametrized closed curves in $S^n$. It is shown that it is not null-homologous (this has some significance for the variational theory of closed geodesics). The construction of the corresponding invariant is reminiscent of the construction of the degree of a map by “smooth means”. This exploits the fact that the homology of $\Pi S^n$ can be constructed using only the singular simplices obtained as follows: in the space of parametrized closed curves, take the singular simplices satisfying some differentiability condition, and project them into $\Pi S^n$ (that is, ignore the orientations and parametrizations of the respective curves).
Bibliography: 13 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1982, 18:3, 403–422

Bibliographic databases:

UDC: 513.83+519.3
MSC: Primary 53C22; Secondary 58B05
Received: 28.01.1981

Citation: D. V. Anosov, “Some homology classes in the space of closed curves in the $n$-dimensional sphere”, Izv. Akad. Nauk SSSR Ser. Mat., 45:3 (1981), 467–490; Math. USSR-Izv., 18:3 (1982), 403–422

Citation in format AMSBIB
\Bibitem{Ano81}
\by D.~V.~Anosov
\paper Some homology classes in the space of closed curves in the $n$-dimensional sphere
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1981
\vol 45
\issue 3
\pages 467--490
\mathnet{http://mi.mathnet.ru/izv2377}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=623347}
\zmath{https://zbmath.org/?q=an:0488.58007|0473.58008}
\transl
\jour Math. USSR-Izv.
\yr 1982
\vol 18
\issue 3
\pages 403--422
\crossref{https://doi.org/10.1070/IM1982v018n03ABEH001392}


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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. A. T. Fomenko, “The topology of surfaces of constant energy in integrable Hamiltonian systems, and obstructions to integrability”, Math. USSR-Izv., 29:3 (1987), 629–658  mathnet  crossref  mathscinet  zmath
    2. I. A. Taimanov, “Closed extremals on two-dimensional manifolds”, Russian Math. Surveys, 47:2 (1992), 163–211  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. M Clapp, “Critical point theory of symmetric functions and closed geodesics”, Differential Geometry and its Applications, 6:4 (1996), 367  crossref
    4. V. I. Arnol'd, A. A. Bolibrukh, R. V. Gamkrelidze, V. P. Maslov, E. F. Mishchenko, S. P. Novikov, Yu. S. Osipov, Ya. G. Sinai, A. M. Stepin, L. D. Faddeev, “Dmitrii Viktorovich Anosov (on his 60th birthday)”, Russian Math. Surveys, 52:2 (1997), 437–445  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    5. I. A. Taimanov, “The type numbers of closed geodesics”, Reg Chaot Dyn, 15:1 (2010), 84  crossref  isi  elib
    6. S. M. Aseev, V. M. Buchstaber, R. I. Grigorchuk, V. Z. Grines, B. M. Gurevich, A. A. Davydov, A. Yu. Zhirov, E. V. Zhuzhoma, M. I. Zelikin, A. B. Katok, A. V. Klimenko, V. V. Kozlov, V. P. Leksin, M. I. Monastyrskii, A. I. Neishtadt, S. P. Novikov, E. A. Sataev, Ya. G. Sinai, A. M. Stepin, “Dmitrii Viktorovich Anosov (obituary)”, Russian Math. Surveys, 70:2 (2015), 369–381  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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