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

TL;DR

This paper constructs non-trivial cocycles for the space of long embeddings using configuration space integrals associated with multi-loop BCR graphs, and applies these to produce explicit non-trivial families of trivial embeddings.

Contribution

It introduces a new method to construct cocycles via configuration space integrals linked to BCR graphs with multiple loops, advancing the understanding of embedding spaces.

Findings

Constructed explicit non-trivial cocycles for long embedding spaces.

Provided a family of trivial embeddings distinguished by cocycle-cycle pairing.

Demonstrated non-triviality using graph and chord diagram pairings.

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.