Journal of Topology

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

For smooth manifolds M and N, let be the homotopy fibre of the inclusion map . Consider the functor from the category of Euclidean spaces to the category of spectra, defined by the 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.

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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.

References

1. Arone G. A note on the homology of _n, the Schwartz genus, and solving polynomial equations. An Alpine anthology of homotopy theory (2006) Providence, RI: American Mathematical Society. 1–10. Contemporary Mathematics 399.
2. Arone G., Kankaanrinta M. The homology of certain subgroups of the symmetric group with coefficients in . J. Pure Appl. Algebra (1998) 127:1–14.[CrossRef]
3. Arone G., Lambrechts P., Voli I. Calculus of functors, operad formality, and rational homology of embedding spaces. Acta Math. (2007) 199:153–198.[CrossRef]
4. Bousfield A., Kan D. Homotopy limits, completions, and localizations (1972) 304. Berlin–New York: Springer. Lecture Notes in Mathematics.
5. Crabb M., James I. Fibrewise homotopy theory (1998) London: Springer. Springer Monographs in Mathematics.
6. Dwyer W., Kan D. Function complexes for diagrams of simplicial sets. Ned. Akad. Wet. Indag. Math. (1983) 45:139–147.
7. Goodwillie T., Klein J., Weiss M. A Haefliger style description of the embedding calculus tower. Topology (2003) 42:509–524.[CrossRef][Web of Science]
8. May J. P., Sigurdsson J. Parametrized homotopy theory (2006) 132. Providence, RI: American Mathematical Society. Mathematical Surveys and Monographs.
9. Weiss M. Orthogonal calculus. Trans. Amer. Math. Soc. (1995) 10:3743–3796.
10. Weiss M. Embeddings from the point of view of immersion theory I. Geom. Topol. (1999) 3:67–101.[CrossRef]

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This Article Abstract Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive Download to citation manager Request Permissions Google Scholar Articles by Arone, G. Search for Related Content Social Bookmarking          What's this?