Derivatives of embedding functors I: the stable case

TL;DR

This paper describes the derivatives of a functor related to embeddings and immersions of manifolds using orthogonal calculus, involving rooted forests and spectra, in the stable case where the target manifold is a tame stably parallelizable manifold.

Contribution

It provides a novel description of the Taylor derivatives of the embedding-immersion homotopy fiber functor using rooted forests and spectra, in the stable case of the target manifold.

Findings

Derivatives are described via spectra of sections over a space of trees.

Construction involves a homotopy bundle of spectra over rooted forests.

First in a series of two papers on derivatives of embedding-related functors.

Abstract

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 $M$.