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

Yoshioka, Leo

doi:10.2140/agt.2025.25.3385

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

Leo Yoshioka

Algebraic & Geometric Topology 25 (2025) 3385–3427

DOI: 10.2140/agt.2025.25.3385

Abstract

The purpose of this paper is to construct nontrivial cocycles of the space $Emb (ℝ^{j}, ℝ^{n})$ of long embeddings. We construct the cocycles through configuration space integrals, associated with Bott–Cattaneo–Rossi graphs with more than one loop. As an application, we give a nontrivial cycle of long embeddings in the path component of the trivial long embedding, for odd $n, j$ with $n - j \geq 2$ and $j \geq 3$ . This cycle is constructed from a chord diagram on directed lines. The nontriviality is shown by cocycle-cycle pairing, described by pairing between graphs and chord diagrams.

Keywords

long embeddings, configuration space integrals, graph complexes

Mathematical Subject Classification

Primary: 57K16, 57K45, 57M15, 57R40

References

Publication

Received: 28 February 2023

Revised: 10 May 2024

Accepted: 17 June 2024

Published: 1 October 2025

Authors

Leo Yoshioka
Graduate School of Mathematical Sciences The University of Tokyo Tokyo Japan

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Volume 25, issue 6 (2025)