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.

arXiv comment52 pages

References in corpus (11)

• Embeddings from the point of view of immersion theory: Part I
• Embeddings from the point of view of immersion theory: Part II
• Configuration spaces and Vassiliev classes in any dimension
• Wilson surfaces and higher dimensional knot invariants
• A family of embedding spaces
• On the rational homology of high dimensional analogues of spaces of long knots
• Nontrivial classes in $H^{\ast }(Imb(S^1,{\mathbb{R}}^n))$ from nontrivalent graph cocycles
• Non-trivalent graph cocycle and cohomology of the long knot space
• Configuration space integrals for embedding spaces and the Haefliger invariant
• An integral expression of the first non-trivial one-cocycle of the space of long knots in R^3
• 1-loop graphs and configuration space integral for embedding spaces