The actions of a circumference on smooth odd-dimensional spheres

The J-invariant of a smooth structure on an odd-dimensional sphere is constructed, which allows us to obtain information on smooth structures on a sphere with free action of a circumference and, in particular, yields complete information when the sphere is of dimension seven.Translated from Matematicheskie Zametki, Vol. 5, No. 4, pp. 401–408, 1969.In conclusion the author considers it to be his duty to express thanks to his scientific mentor S. P. Novikov for posing the problem and helping with the work, and also to V. M. Bykhshtaber for valuable discussions.