Configuration space integrals for embedding spaces and the Haefliger invariant
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.
Cite
@article{arxiv.0811.3726,
title = {Configuration space integrals for embedding spaces and the Haefliger invariant},
author = {Keiichi Sakai},
journal= {arXiv preprint arXiv:0811.3726},
year = {2011}
}
Comments
41 pages, many figures (v2: Theorem 1.3 and its proof have been improved. many minor corrections. v3: Theorem 1.3 and its proof have been revised, since Lemma 5.26 in v2 was wrong. v4: The proof of Theorem 1.3 has been fully revised. To appear in J. Knot Theory Ramifications)