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Journal of Topology Advance Access originally published online on August 17, 2009
Journal of Topology 2009 2(3):461-516; doi:10.1112/jtopol/jtp019
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© 2009 London Mathematical Society

Derivatives of embedding functors I: the stable case

Gregory Arone

Department of Mathematics, University of Virginia, Charlottesville, Virginia, USA zga2m@virginia.edu


  Abstract

For smooth manifolds M and N, let Formula be the homotopy fibre of the inclusion map Formula . Consider the functor from the category of Euclidean spaces to the category of spectra, defined by the formula Formula . In this paper, we describe the derivatives of this functor, in the sense of M. Weiss's orthogonal calculus, in the case when N is a stably parallelizable manifold (we believe that the parallelizability assumption is not essential). Our construction involves a certain space of partitions of M (or, equivalently, a space of rooted forests with leaves marked by points in M), and a certain `homotopy bundle of spectra' over this space of trees. The nth derivative is then described as the `spectrum of restricted sections' of this bundle.

Received May 16, 2008.

2000 Mathematics Subject Classification 57N35

The author gratefully acknowledges that he was supported by the National Science Foundation for the preparation of this paper, most recently via grant no. DMS 0605073.


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