Cyclic objects from surfaces

In this paper, we endow the family of closed oriented genus $g$ surfaces, starting with torus, with a structure of a (co)cyclic object in the category of $3$-dimensional cobordisms. As a corollary, any $3$-dimensional TQFT induces a (co)cyclic module, which we compute algebraically for the Reshetikhin-Turaev TQFT.
48 pages, many figures. Comments welcome!