Loop structures in Taylor towers

Abstract

We study spaces of natural transformations between homogeneous functors in Goodwillie’s calculus of homotopy functors and in Weiss’s orthogonal calculus. We give a description of such spaces of natural transformations in terms of the homotopy fixed point construction. Our main application uses this description in combination with the Segal Conjecture to obtain a delooping theorem for connecting maps in the Goodwillie tower of the identity and in the Weiss tower of BU.V/. The interest in such deloopings stems from conjectures made by the first and the third author [4] that these towers provide a source of contracting homotopies for certain projective chain complexes of spectra. 55P65; 55P47, 18G55 1 Introduction and notation In this paper, we study homogeneous functors in the sense of Weiss’s orthogonal calculus [13]. More precisely, we calculate the space of natural transformations between such functors, and we give a few examples and applications. Our main application is a delooping theorem for connecting maps in the Goodwillie tower of the identity functor for pointed spaces and in the Weiss tower of the functor V7!BU.V/. The motivation for this delooping theorem will be discussed later in this introduction. To state our results, we summarize the usual notation for this context. Let F be a functor from pointed spaces or from finite-dimensional real or complex vector spaces to pointed spaces or spectra. Goodwillie and Weiss calculus assign to such a functor F a “Taylor” tower of functors ! Pn F! Pn 1 F! ; together with a natural map from F to the homotopy inverse limit of the tower that is often a weak homotopy equivalence. The homotopy fiber of the map Pn F! Pn 1 F is customarily denoted Dn F and is referred to as “the nth homogeneous layer of F ,” while Pn F is referred to as “the nth Taylor polynomial of F .” Depending on whether