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Cocycles of the space of long embeddings and BCR graphs with more than one loop

Leo Yoshioka

TL;DR

The paper constructs nontrivial cocycles of the space Emb(R^j, R^n) using configuration space integrals associated with higher-loop Bott-Cattaneo-Rossi graphs. It introduces a 2-loop BCR graph cocycle H of order 3, builds corresponding generalized ribbon cycles from chord diagrams on directed lines, and develops correction terms to handle anomalous faces, enabling a cocycle–cycle pairing that proves nontriviality. For odd n and j with n-j >= 2 and j >= 3, the resulting cocycle yields a nontrivial class in the de Rham cohomology of Emb(R^j, R^n) (mod immersions), and an explicit S^2-family of trivial long 3-knots; these constructions align with, and extend, Arone-Turchin type graph complexes. Overall, the work broadens the BCR graph framework to higher loops, providing computable pairings and insights into stability phenomena in embedding spaces at codimension two.

Abstract

The purpose of this paper is to construct non-trivial cocycles of the space $Emb(\mathbb{R}^j, \mathbb{R}^{n})$ of long embeddings. We construct the cocycles by integral over configuration spaces, associated with Bott-Cattaneo-Rossi graphs with more than one loop. As an application, we give explicitly a non-trivial family of trivial long embeddings for odd $n,j$ with $n-j \geq 2$ and $j \geq 3$. This family (cycle) is constructed from a chord diagram on directed lines. The non-triviality is shown by cocycle-cycle paring, described by paring between graphs and chord diagrams.

Cocycles of the space of long embeddings and BCR graphs with more than one loop

TL;DR

The paper constructs nontrivial cocycles of the space Emb(R^j, R^n) using configuration space integrals associated with higher-loop Bott-Cattaneo-Rossi graphs. It introduces a 2-loop BCR graph cocycle H of order 3, builds corresponding generalized ribbon cycles from chord diagrams on directed lines, and develops correction terms to handle anomalous faces, enabling a cocycle–cycle pairing that proves nontriviality. For odd n and j with n-j >= 2 and j >= 3, the resulting cocycle yields a nontrivial class in the de Rham cohomology of Emb(R^j, R^n) (mod immersions), and an explicit S^2-family of trivial long 3-knots; these constructions align with, and extend, Arone-Turchin type graph complexes. Overall, the work broadens the BCR graph framework to higher loops, providing computable pairings and insights into stability phenomena in embedding spaces at codimension two.

Abstract

The purpose of this paper is to construct non-trivial cocycles of the space of long embeddings. We construct the cocycles by integral over configuration spaces, associated with Bott-Cattaneo-Rossi graphs with more than one loop. As an application, we give explicitly a non-trivial family of trivial long embeddings for odd with and . This family (cycle) is constructed from a chord diagram on directed lines. The non-triviality is shown by cocycle-cycle paring, described by paring between graphs and chord diagrams.
Paper Structure (28 sections, 28 theorems, 70 equations, 19 figures)

This paper contains 28 sections, 28 theorems, 70 equations, 19 figures.

Table of Contents

  1. Background
  2. The BCR graph complex and the de Rham complex of $\text{Emb}(\mathbb{R}^j, \mathbb{R}^n)$
  3. Definition
  4. The space of long embeddings
  5. BCR graphs
  6. The BCR graph complex
  7. Comparing with the Arone-Turchin graph complex
  8. Configuration space integrals
  9. Main Result
  10. Construction of the cocycles
  11. H is a graph cocycle
  12. Vanishing of contributions of hidden faces
  13. Construction of correction terms for anomalous faces
  14. Review of Sakai and Watanabe's construction of the wheel-like cycles
  15. The wheel-like ribbon presentations
  16. ...and 13 more sections

Key Result

Theorem 1

Let $n$, $j$ be odd and satisfy $n-j\geq 2$, $j\geq 3$. Then the linear combination $H$ of $2$-loop BCR graphs gives an explicit non-trivial $3n-2j-7$ (co)cycle of $\overline{\text{Emb}}(\mathbb{R}^j, \mathbb{R}^{n})$, the space of long embeddings modulo immersions.

Figures (19)

  • Figure 1: Graph and its direction maps
  • Figure 2: Non-trivial graph cocycle $H$ (odd, odd)
  • Figure 3: An example of the matrix of $\delta$. For our cocycle $H$, the blocks with $\ast$ are $0$. The blocks with $\star$ do not appear.
  • Figure 4: Perturbation of a crossing
  • Figure 5: $\varphi^\mathbf{v, \varepsilon}_3$ as embeddings from standard disks and annuli
  • ...and 14 more figures

Theorems & Definitions (100)

  • Theorem 1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.6
  • ...and 90 more