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A model for framed configuration spaces of points

Ricardo Campos, Julien Ducoulombier, Najib Idrissi, Thomas Willwacher

TL;DR

This work develops real homotopy descriptions for framed configuration spaces on closed oriented manifolds by introducing explicit graphical (graph) models that capture the action of the framed little discs operad. A fiberwise variant $\mathsf{FM}_{n}^{M}$ and a framing construction yield a framed configuration module $\mathsf{FM}_{M}^{\mathrm{fr}}$, alongside graphical models ${\mathsf{Graphs}}_{M}^{\mathrm{fr}}$ and ${\mathsf{Graphs}}_{n}^{\mathrm{fr}}$ that coherently model the operadic coaction and the $\mathrm{SO}(n)$-equivariant structure; a zigzag with PA-forms links these combinatorial objects to geometry. The authors prove that the graphical zigzags are quasi-isomorphisms compatible with the framed operad actions, providing computable real (and rational) descriptions of the framed configuration spaces and their embedding-space applications, including behavior under framing changes for parallelizable manifolds. The framework thereby enables explicit, computable control of embedding spaces and factorization homology in the framed setting, extending the Goodwillie–Weiss-type analyses to closed oriented manifolds through graphical and homotopical tools.

Abstract

We study configuration spaces of framed points on oriented closed smooth manifolds. Such configuration spaces admit natural actions of the framed little discs operads, that play an important role in the study of embedding spaces of manifolds and in factorization homology. We construct real combinatorial models for these operadic modules, for orientable closed smooth manifolds.

A model for framed configuration spaces of points

TL;DR

This work develops real homotopy descriptions for framed configuration spaces on closed oriented manifolds by introducing explicit graphical (graph) models that capture the action of the framed little discs operad. A fiberwise variant and a framing construction yield a framed configuration module , alongside graphical models and that coherently model the operadic coaction and the -equivariant structure; a zigzag with PA-forms links these combinatorial objects to geometry. The authors prove that the graphical zigzags are quasi-isomorphisms compatible with the framed operad actions, providing computable real (and rational) descriptions of the framed configuration spaces and their embedding-space applications, including behavior under framing changes for parallelizable manifolds. The framework thereby enables explicit, computable control of embedding spaces and factorization homology in the framed setting, extending the Goodwillie–Weiss-type analyses to closed oriented manifolds through graphical and homotopical tools.

Abstract

We study configuration spaces of framed points on oriented closed smooth manifolds. Such configuration spaces admit natural actions of the framed little discs operads, that play an important role in the study of embedding spaces of manifolds and in factorization homology. We construct real combinatorial models for these operadic modules, for orientable closed smooth manifolds.

Paper Structure

This paper contains 31 sections, 18 theorems, 167 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Background and recollections
  4. The Fulton-MacPherson-Axelrod-Singer compactifications
  5. Homotopy (co)operads and (co)modules, rational homotopy theory of operads
  6. Formality of $\mathsf{FM}_{n}$
  7. The Kontsevich graph complex $\mathrm{GC}_n$
  8. Graphical models for $\mathsf{FM}_M$
  9. Equivariant graphical models
  10. The fiber-wise little discs operad
  11. Motivation
  12. Definition
  13. A model for the frame bundle
  14. Graphical model for the fiber-wise little discs operad
  15. The framed configuration module
  16. ...and 16 more sections

Key Result

Theorem 1

The zigzag ${\mathsf{Graphs}}_M^{\mathrm{fr}}(k) \gets \cdot \to \mathop{\mathrm{\Omega_{\mathrm{PA}}}}\nolimits(\mathsf{FM}^{\mathrm{fr}}_M(k))$ is a weak equivalence, and it is compatible with the action of $\mathsf{FM}^{\mathrm{fr}}_{m}$ on $\mathsf{FM}^{\mathrm{fr}}_{M}$.

Figures (4)

  • Figure 1: Illustration of the operadic structure of $\mathsf{FM}_{n}$.
  • Figure 2: Illustration of the right $\mathsf{FM}_{n}$-module structure on $\mathsf{FM}_{M}$.
  • Figure 3: Illustration of an element in ${\mathsf{Graphs}}_{n}(4)$ with $3$ internal vertices.
  • Figure 4: Illustration of an element in ${\mathsf{Graphs}}_{M}(4)$.

Theorems & Definitions (46)

  • Theorem 1: See Theorem \ref{['thm:model-ffmm']}
  • Theorem 2: See Theorem \ref{['thm:graphs-m-model']}
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9
  • Theorem 10: Kontsevich1999LambrechtsVolic2014
  • ...and 36 more