Power maps and epicyclic spaces

AbstractIn this paper, we present the basic facts of the theory of epicyclic spaces for the first time considered by T. Goodwillie in an unpublished letter to F. Waldhausen. An epicyclic space is a space-valued, contravariant functor on a category ʌ̃ which contains the cyclic category. Our results parallel basic facts of the theory of cyclic spaces established in [1] and [7]. We show that the geometric realization of an epicyclic space has an action of a monoid which is a semidirect product of S1 and the multiplicative monoid of natural numbers. We also show that the homotopy colimit of an epicyclic space is homotopy equivalent to the bar construction for the monoid action. Finally, we give an explicit description of the homotopy type of the classifying space of the category ʌ̃