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

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 $n-j\ge 2$ and $j\ge 3$. 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.

52 pages

References in corpus (2)

• A family of embedding spaces
• An integral expression of the first non-trivial one-cocycle of the space of long knots in R^3