Loop structures in Taylor towers

Arone, Gregory Z¹ ; Dwyer, William G² ; Lesh, Kathryn³

¹Kerchof Hall, University of Virginia, PO Box 400137, Charlottesville VA 22904, USA
²Department of Mathematics, University of Notre Dame, Notre Dame IN 46556, USA
³Department of Mathematics, Union College, Schenectady NY 12309, USA

Algebraic and Geometric Topology, Tome 8 (2008) no. 1, pp. 173-210

Résumé

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.

Détail

DOI : 10.2140/agt.2008.8.173

Keywords: derived natural transformations, Whitehead Conjecture, homotopy calculus, orthogonal calculus, homogeneous functors, delooping, Segal Conjecture

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