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Some non-trivial cycles of the space of long embeddings detected by configuration space integral invariants using g-loop (g =2, 3) graphs

Leo Yoshioka

TL;DR

This work constructs explicit geometric cycles in the space of long embeddings modulo immersions $ar{ ext{K}}_{n,j}$ by exploiting decorated graph complexes and modified configuration space integrals. It develops 2-loop and 3-loop cycle families via chord diagrams $D( ext{Θ}(p,q,r))$, corresponding ribbon presentations $P( ext{Θ}(p,q,r))$, and their perturbations, proving their non-triviality through a counting formula that pairs cycles with graph cocycles. The results yield infinite-dimensional rational homotopy groups of embedding spaces, extending earlier 1-loop findings to higher-loop sensitivity and connecting to known diffeomorphism group phenomena; the framework also suggests links to claspers and higher-dimensional analogs. Overall, the paper provides a robust geometric-d combinatorial method to detect non-finite generation phenomena in spaces of long embeddings in codimension $ eq 1$, using 2-loop and 3-loop hairy graphs as the driving invariants.

Abstract

In this paper, we give some non-trivial geometric cycles of the space of long embeddings R^j --> R^n (n-j >= 2) modulo immersions. We construct a class of cycles from specific chord diagrams associated with the 2-loop or 3-loop hairy graphs. To detect these cycles, we use cocycles obtained by the 2-loop or 3-loop part of modified configuration space integrals using a modified Bott-Cattaneo-Rossi graph complex. We show the non-triviality of the cycles by pairing argument, which is reduced to pairing of graphs with the chord diagrams. As a corollary of the 2-loop part, we provide an alternative proof of the non-finite generation of the (j-1)-th rational homotopy group of the space of long embeddings of codimension two, which Budney--Gabai and Watanabe first established. We also show the non-finite generation of the 2(j-1)-th homotopy group by using the 3-loop part.

Some non-trivial cycles of the space of long embeddings detected by configuration space integral invariants using g-loop (g =2, 3) graphs

TL;DR

This work constructs explicit geometric cycles in the space of long embeddings modulo immersions by exploiting decorated graph complexes and modified configuration space integrals. It develops 2-loop and 3-loop cycle families via chord diagrams , corresponding ribbon presentations , and their perturbations, proving their non-triviality through a counting formula that pairs cycles with graph cocycles. The results yield infinite-dimensional rational homotopy groups of embedding spaces, extending earlier 1-loop findings to higher-loop sensitivity and connecting to known diffeomorphism group phenomena; the framework also suggests links to claspers and higher-dimensional analogs. Overall, the paper provides a robust geometric-d combinatorial method to detect non-finite generation phenomena in spaces of long embeddings in codimension , using 2-loop and 3-loop hairy graphs as the driving invariants.

Abstract

In this paper, we give some non-trivial geometric cycles of the space of long embeddings R^j --> R^n (n-j >= 2) modulo immersions. We construct a class of cycles from specific chord diagrams associated with the 2-loop or 3-loop hairy graphs. To detect these cycles, we use cocycles obtained by the 2-loop or 3-loop part of modified configuration space integrals using a modified Bott-Cattaneo-Rossi graph complex. We show the non-triviality of the cycles by pairing argument, which is reduced to pairing of graphs with the chord diagrams. As a corollary of the 2-loop part, we provide an alternative proof of the non-finite generation of the (j-1)-th rational homotopy group of the space of long embeddings of codimension two, which Budney--Gabai and Watanabe first established. We also show the non-finite generation of the 2(j-1)-th homotopy group by using the 3-loop part.

Paper Structure

This paper contains 18 sections, 22 theorems, 58 equations, 18 figures.

Table of Contents

  1. Preliminaries for construction of cycles
  2. The space of long embeddings
  3. Ribbon presentations and long embeddigs associated with them
  4. Cycles of $\overline{\mathcal{K}}_{n,j}$ obtained by perturbation of crossings of ribbon presentations
  5. Planetary systems and chord diagrams on oriented lines
  6. Construction of $2$-loop and $3$-loop cycles of $\overline{\mathcal{K}}_{n,j}$
  7. The chord diagrams $D(\Theta(p,q,r))$
  8. The ribbon presentations $P(\Theta(p,q,r))$
  9. The $2$-loop cycles $d(\Theta(p,q,r))$ of $\overline{\mathcal{K}}_{n,j}$
  10. Properties of the ribbon presentations $P(\Theta(p,q,r))$
  11. The class $\mathcal{H}_{n,j}(g=2)$ of $2$-loop cycles
  12. The class $\mathcal{H}_{n,j}(g=3)$ of $3$-loop cycles
  13. Non-triviality of $2$-loop and $3$-loop cycles of $\overline{\mathcal{K}}_{n,j}$
  14. Review of the modified configuration space integrals
  15. The counting formula of configuration space integrals
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\mathcal{H}_{n,j}(g)$ ($g=2, 3$) be the subspace of $H_{\ast}(\overline{\mathcal{K}}_{n,j})$ which is generated by (replaced) $g$-loop cycles in Section Construction of general $2$-loop and $3$-loop cycles. Then there is a map which is non-degenerate with respect to $H^{top}(HGC_{n,j}(g))$. Here, $H^{top}(HGC_{n,j})$ is the top cohomology of $HGC_{n,j}$The top cohomology is the subspace of t

Figures (18)

  • Figure 1: Example of moves of ribbon presentations
  • Figure 2: The hairy graph $\Theta(p,q,r)$
  • Figure 3: The diagram $D(\Theta(4,3,2))$)
  • Figure 4: The set of planetary systems corresponding to $\Theta(p,q,r)$
  • Figure 5: The ribbon presentation $P(\Theta(4,3,2))$)
  • ...and 13 more figures

Theorems & Definitions (58)

  • Theorem 1: Theorem \ref{['nontrivialityof2loopcycles']}, \ref{['nontrivialityof3loopcycles']}
  • Corollary 2
  • Definition 1.1
  • Definition 1.2
  • Definition 1.4: Ribbon presenations HKSHS
  • Definition 1.9
  • Definition 1.10
  • Remark 1.11
  • Definition 1.14
  • Definition 1.16
  • ...and 48 more