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

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 [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.

derived natural transformations, Whitehead Conjecture, homotopy calculus, orthogonal calculus, homogeneous functors, delooping, Segal Conjecture
Mathematical Subject Classification 2000
Primary: 55P65
Secondary: 55P47, 18G55
Received: 24 April 2007
Revised: 30 November 2007
Accepted: 19 December 2007
Published: 8 February 2008
Gregory Z Arone
Kerchof Hall
University of Virginia
PO Box 400137
Charlottesville VA 22904
William G Dwyer
Department of Mathematics
University of Notre Dame
Notre Dame IN 46556
Kathryn Lesh
Department of Mathematics
Union College
Schenectady NY 12309