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 $BU\left(V\right)$. The interest in such deloopings stems from conjectures made by the first and the third author [Filtered spectra arising from permutative categories, J. Reine Angew. Math. 604 (2007) 73-136] that these towers provide a source of contracting homotopies for certain projective chain complexes of spectra.