Configuration space integrals for embedding spaces and the Haefliger invariant

Details Configuration space integrals for embedding spaces and the Haefliger invariant Configuration space integrals for embedding spaces and the Haefliger invariant

2008-11-23

arxiv.org/abs/0811.3726v4

Journal of Knot Theory and Its Ramifications, Vol. 19, No. 12 (2010), 1597-1644

Mathematics

Geometric Topology

41 pages, many figures (v2: Theorem 1.3 and its proof have been improved. many minor corrections. v3: Theorem 1.3 and its proo

Scientific paper

10.1142/S0218216510008583

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.

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