Derivatives of embedding functors I: the stable case

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 $M$.

💡 Research Summary

The paper studies the homotopy‑theoretic difference between the space of smooth embeddings (\Emb(M,N)) and the space of immersions (\Imm(M,N)) for smooth manifolds (M) and (N). The authors consider the homotopy fibre (\Ebar(M,N)) of the inclusion (\Emb(M,N)\to\Imm(M,N)) and suspend it to obtain a spectrum (\Sigma^{\infty}\Ebar(M,N)). The main object of study is the functor \