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On hidden face contributions of configuration space integrals for long embeddings

Leo Yoshioka

TL;DR

The paper addresses obstructions to configuration space integrals forming a cochain map for long embeddings by introducing a decorated graph complex $DGC_{n,j}$ whose external vertices carry decorations from an acyclic bar complex $Z=A_{n,j}\otimes_{\tau} BA_{n,j}$. It proves $DGC_{n,j}$ is quasi-isomorphic to the plain graph complex $PGC_{n,j}$ and to Arone–Turchin’s Hochschild and hairy graph complexes, enabling the construction of a genuine cochain map to $\Omega^{*}_{dR}(\overline{\mathcal{K}}_{n,j})$ via Chen’s iterated integrals. The work provides a thorough analysis of degree-zero elements, top cohomology in low genus, and shows how the modified configuration space integrals $\overline{I}$ cancel hidden-face contributions, yielding new cohomology classes for long embeddings modulo immersions. By connecting embedding calculus with decorated graph complexes and iterated integrals, the results offer robust tools for understanding the rational homotopy type of the space of long embeddings and its relations to configuration-space formality.

Abstract

Configuration space integrals are powerful tools for studying the homotopy type of the space of long embeddings in terms of a combinatorial object called a graph complex. It is unknown whether these integrals give a cochain map due to potential obstructions called hidden faces. The purpose of this paper is to address these hidden faces by modifying configuration space integrals: we incorporate the acyclic bar complex of some dg algebra into the original graph complex, without changing its cohomology. Then, we give a cochain map from the new graph complex to the de Rham complex of the space of long embeddings modulo immersions, by combining the original configuration space integrals with Chen's iterated integrals. As the original complex, we choose quite a modified graph complex so that it is quasi-isomorphic to both the hairy graph complex and a graph complex introduced in the context of embedding calculus.

On hidden face contributions of configuration space integrals for long embeddings

TL;DR

The paper addresses obstructions to configuration space integrals forming a cochain map for long embeddings by introducing a decorated graph complex whose external vertices carry decorations from an acyclic bar complex . It proves is quasi-isomorphic to the plain graph complex and to Arone–Turchin’s Hochschild and hairy graph complexes, enabling the construction of a genuine cochain map to via Chen’s iterated integrals. The work provides a thorough analysis of degree-zero elements, top cohomology in low genus, and shows how the modified configuration space integrals cancel hidden-face contributions, yielding new cohomology classes for long embeddings modulo immersions. By connecting embedding calculus with decorated graph complexes and iterated integrals, the results offer robust tools for understanding the rational homotopy type of the space of long embeddings and its relations to configuration-space formality.

Abstract

Configuration space integrals are powerful tools for studying the homotopy type of the space of long embeddings in terms of a combinatorial object called a graph complex. It is unknown whether these integrals give a cochain map due to potential obstructions called hidden faces. The purpose of this paper is to address these hidden faces by modifying configuration space integrals: we incorporate the acyclic bar complex of some dg algebra into the original graph complex, without changing its cohomology. Then, we give a cochain map from the new graph complex to the de Rham complex of the space of long embeddings modulo immersions, by combining the original configuration space integrals with Chen's iterated integrals. As the original complex, we choose quite a modified graph complex so that it is quasi-isomorphic to both the hairy graph complex and a graph complex introduced in the context of embedding calculus.

Paper Structure

This paper contains 28 sections, 54 theorems, 218 equations, 21 figures.

Table of Contents

  1. Preliminaries for the decorated graph complex
  2. The space of long embeddings
  3. Bar complexes and Chen's iterated integrals
  4. Configuration spaces and their cohomology
  5. The graph cooperad $\text{Graph}_n(\bullet)$
  6. The plain graph complex and its operadic description
  7. The plain graph complex $PGC_{n,j}$
  8. Arone--Turchin's Hochschild-type complex $HH_{n,j}$
  9. The hairy graph complex $HGC_{n,j}$
  10. Notation on orientations and signs
  11. Operadic descriptions of $PGC_{n,j}$, $HH_{n,j}$ and $HGC_{n,j}$
  12. Relationship between $PGC_{n,j}$, $HH_{n,j}$ and $HGC_{n,j}$
  13. Remaks on the map at the top cohomology $H^{top}(PGC^{\prime}_{n,j}) \rightarrow H^{top}(HGC_{n,j}$)
  14. Review of computation of the top homologies for $g=2$ and $g=3$
  15. The hidden face dg algebra and the decorated graph complex
  16. ...and 13 more sections

Key Result

Theorem 1

When $n-j \geq 2$ and $j \geq 3$, there exists a cochain map It holds even when $j = 2$, if $\overline{I}$ is restricted to the subspace of $DGC_{n,j}$ generated by decorated graphs with the first Betti number $g \leq 3$.

Figures (21)

  • Figure 1: Overview ($j \geq 2$). The part drawn in red is the scope of this paper.
  • Figure 2: Overview ($j =1$)
  • Figure 3: An element of $H^{\ast}(\text{Conf}_6(\mathbb{R}^n))$
  • Figure 4: Example of the differential of a graph
  • Figure 5: Example of the product of a graph
  • ...and 16 more figures

Theorems & Definitions (153)

  • Theorem 1: Theorem \ref{['main theorem 1 on hidden faces2']}
  • Theorem 2: Theorem \ref{['maintheoremrestate']}. Suggested by Turchin
  • Theorem 3: Theorem \ref{['relationwithAroneTurchincomplex']}
  • Theorem 4: Theorem \ref{['relationwithhairygraphcomplex']}
  • Corollary 5
  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Definition 1.5: Configuration spaces
  • ...and 143 more