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Embedding calculus for parallelized manifolds

Semyon Abramyan

Abstract

We study a variant of the embedding functor $\mathop{\mathrm{Emb}}(M, N)$ that incorporates homotopical data from the frame bundle of the target manifold $N$. Given a parallelized $m$-manifold $M$ and an $n$-manifold $N$ equipped with a section of its $m$-frame bundle, we define a modified embedding functor $\widetilde{\mathop{\mathrm{Emb}}}(M, N)$ that interpolates between the standard embedding and a reference framing. Using the manifold calculus of functors, we identify the Taylor tower of $\widetilde{\mathop{\mathrm{Emb}}}(M, N)$ with a mapping space of right modules over the Fulton-MacPherson operad. We prove a convergence theorem under a codimension condition, establishing a weak equivalence between $\widetilde{\mathop{\mathrm{Emb}}}(M, N)$ and its Taylor approximation. Finally, under rationalization, we describe the derived mapping space in terms of a combinatorial hairy graph complex, enabling computational access to the rational homotopy type of the space of embeddings.

Embedding calculus for parallelized manifolds

Abstract

We study a variant of the embedding functor that incorporates homotopical data from the frame bundle of the target manifold . Given a parallelized -manifold and an -manifold equipped with a section of its -frame bundle, we define a modified embedding functor that interpolates between the standard embedding and a reference framing. Using the manifold calculus of functors, we identify the Taylor tower of with a mapping space of right modules over the Fulton-MacPherson operad. We prove a convergence theorem under a codimension condition, establishing a weak equivalence between and its Taylor approximation. Finally, under rationalization, we describe the derived mapping space in terms of a combinatorial hairy graph complex, enabling computational access to the rational homotopy type of the space of embeddings.

Paper Structure

This paper contains 21 sections, 15 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topological $W$-construction
  4. Versions of the Fulton-MacPherson operad
  5. Model structure on modules over an operad
  6. Algebraic $W$-construction
  7. Graph complexes and graph operads
  8. The $\mathsf{Graphs}_n$-comodule $\mathsf{Graphs}_{V, n}$
  9. Hairy graph complexes
  10. Operadic part
  11. Setting up
  12. The limit of the Taylor tower for $\widetilde{\mathop{\mathrm{Emb}}\nolimits}(M, N)$
  13. Convergence. Proof of \ref{['main-convergence']}
  14. Passing to graph complexes. Proof of \ref{['main-computation']}
  15. Motivation
  16. ...and 6 more sections

Key Result

Proposition 2.2

Let $\mathcal{P}$ be a topological operad such that $\{\iota\} \hookrightarrow \mathcal{P}(1)$ is a cofibration and each $\mathcal{P}(n)$ is a cofibrant $S_n$-space. Then $W\mathcal{P}$ is a cofibrant resolution of $\mathcal{P}$ with the map $W\mathcal{P} \xrightarrow{\sim} \mathcal{P}$ contracting

Theorems & Definitions (17)

  • Definition 1.1: $\widetilde{\mathop{\mathrm{Emb}}\nolimits}(M, N)$
  • Proposition 2.2: boar71
  • Proposition 2.4: salv01
  • Theorem 2.5: see fres09
  • Proposition 2.7: f-t-w17
  • Proposition 2.9
  • Theorem 2.10: Kontsevich, Lambrechts-Volić
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 3.3
  • ...and 7 more