Volume 8, issue 1 (2008)

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Loop structures in Taylor towers

Gregory Z Arone, William G Dwyer and Kathryn Lesh

Algebraic & Geometric Topology 8 (2008) 173–210
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.

Keywords
derived natural transformations, Whitehead Conjecture, homotopy calculus, orthogonal calculus, homogeneous functors, delooping, Segal Conjecture
Mathematical Subject Classification 2000
Primary: 55P65
Secondary: 55P47, 18G55