Non-trivalent graph cocycle and cohomology of the long knot space

Abstract

In this paper we show that via the configuration space integral construction a non-trivalent graph cocycle can also yield a non-zero cohomology class of the space of higher (and even) codimensional long knots. This simultaneously proves that the Browder operation induced by the operad action defined by R. Budney is not trivial.

17 pages, 11 figures (v2: a comment on a work of R. Longoni is added. v3: Remark 3.6 of v2 has been removed since it might be wrong, as pointed out by I. Volic. We work on R^n instead of the cylinder. Sections 2.2 and 3.3 have been widely revised. Many other minor revisions and corrections.)

References in corpus (3)

• Configuration spaces and Vassiliev classes in any dimension
• Nontrivial classes in $H^{\ast }(Imb(S^1,{\mathbb{R}}^n))$ from nontrivalent graph cocycles
• Poisson structures on the homology of the space of knots

Cited by in corpus (8)

• Configuration space integrals for embedding spaces and the Haefliger invariant
• A homotopy-theoretic view of Bott-Taubes integrals and knot spaces
• An integral expression of the first non-trivial one-cocycle of the space of long knots in R^3
• Knotted families from graspers
• BV-structures on the homology of the framed long knot space
• A Kontsevich integral of order 1
• The Fox-Hatcher cycle and a Vassiliev invariant of order three
• Cocycles of the space of long embeddings and BCR graphs with more than one loop