Configuration space integrals for embedding spaces and the Haefliger invariant

TL;DR

This paper develops a graph complex and configuration space integrals to study embedding spaces, providing new cocycles and a novel formulation of the Haefliger invariant in certain dimensions.

Contribution

Introduces a graph complex and integral map that produce cocycles and a new formulation of the Haefliger invariant for long knots in specific dimension ranges.

Findings

Map I is a cochain map under certain conditions on graphs and dimensions.

Constructs new cocycles related to the Haefliger invariant.

Provides a framework connecting graph complexes, integrals, and embedding space invariants.

Abstract

Let K be the space of long j-knots in R^n. In this paper we introduce a graph complex D and a linear map I from D to the de Rham complex of K via configuration space integral, and prove that (1) when both n>j>=3 are odd, the map I is a cochain map if restricted to graphs with at most one loop component, (2) when n-j>=2 is even, the map I is a cochain map if restricted to tree graphs, and (3) when n-j >=3 is odd, the map I added a correction term produces a (2n-3j-3)-cocycle of K which gives a new formulation of the Haefliger invariant when n=6k, j=4k-1 for some k.