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Some homology classes in the space of closed curves in the $n$-dimensional sphere
D. V. Anosov
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).
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Mathematics of the USSR-Izvestiya, 1982, 18:3, 403–422
MSC: Primary 53C22; Secondary 58B05
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
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\paper Some homology classes in the space of closed curves in the $n$-dimensional sphere
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\jour Math. USSR-Izv.
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