SOME HOMOLOGY CLASSES IN THE SPACE OF CLOSED CURVES IN THEn-DIMENSIONAL SPHERE

Abstract

The (n–1)-dimensional mod 2 cycle generated by the great circles passing through two fixed, diametrically opposite points in the n-dimensional sphere Sn is considered in the space ΠSn of nonoriented, nonparametrized closed curves in Sn. 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 ΠSn 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 ΠSn (that is, ignore the orientations and parametrizations of the respective curves). Bibliography: 13 titles.