## DERIVATIVES OF EMBEDDING FUNCTORS I: THE STABLE CASE (2007)

### BibTeX

@MISC{Arone07derivativesof,

author = {Gregory Arone},

title = {DERIVATIVES OF EMBEDDING FUNCTORS I: THE STABLE CASE},

year = {2007}

}

### OpenURL

### Abstract

For smooth manifolds M and N, let Emb(M, N) be the homotopy fiber of the map Emb(M, N) − → Imm(M, N). Consider the functor from the category of Euclidean spaces to the category of spectra, defined by the formula V ↦ → Σ ∞ Emb(M, N × V). In this paper, we describe the derivatives of this functor, in the sense of M. Weiss ’ orthogonal calculus. Our construction involves a certain space of rooted forests (or, equivalently, a space of partitions) with leaves marked by points in M, and a certain “homotopy bundle of spectra ” over this space of trees. The n-th derivative is then described as the “spectrum of restricted sections ” of this bundle. This is the first in a series of two papers. In the second part, we will give an analogous description of the derivatives of the functor Emb(M, N × V), involving a similar construction with certain spaces