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

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.