MANIFOLDS, K-THEORY AND THE CALCULUS OF FUNCTORS

Abstract. The Taylor tower of a functor from based spaces to spectra can be classified according to the action of a certain comonad on the collection of derivatives of the functor. We describe various equivalent conditions under which this action can be lifted to the structure of a module over the Koszul dual of the little L-discs operad. In particular, we show that this is the case when the functor is a left Kan extension from a certain category of ‘pointed framed L-manifolds ’ and pointed framed embeddings. As an application we prove that the Taylor tower of Waldhausen’s algebraic K-theory of spaces functor is classified by an action of the Koszul dual of the little 3-discs operad. In previous work [2, 3] we described the structure possessed by the Goodwillie derivatives of