## Derivatives of embedding functors I: the stable case

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Arone, Gregory

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For smooth manifolds $M$ and $N$, let $\Ebar(M, N)$ be the homotopy fiber of
the map $\Emb(M, N)\longrightarrow \Imm(M, N)$. Consider the functor from the
category of Euclidean spaces to the category of spectra, defined by the formula
$V\mapsto \Sigma^\infty\Ebar(M, N\times V)$. In this paper, we describe the
Taylor polynomials of this functor, in the sense of M. Weiss' orthogonal
calculus, in the case when $N$ is a nice open submanifold of a Euclidean space.
This leads to a description of the derivatives of this functor when $N$ is a
tame stably parallelizable manifold (we believe that the parallelizability
assumption is not essential). 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 $\Ebar(M, N\times V)$, involving a similar construction with certain
spaces of connected graphs (instead of forests) with points marked in $M$.

Comment: 58 pages, 1 figure. This is a major rewrite of the previously posted version. The proof has been thoroughly reorganized, and we hope it has become simpler and more direct (the new version is 17 pages shorter than the previous one)

Comment: 58 pages, 1 figure. This is a major rewrite of the previously posted version. The proof has been thoroughly reorganized, and we hope it has become simpler and more direct (the new version is 17 pages shorter than the previous one)

##### Keywords

Mathematics - Algebraic Topology, 57N35