Nontrivalent graph cocycle and cohomology of the long knot space

Abstract

1499Nontrivalent graph cocycle and cohomology of the long knot space KEIICHI SAKAIIn this paper we show that via the configuration space integral construction a nontrivalent graph cocycle can also yield a nonzero 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.

A homotopy-theoretic view of Bott–Taubes integrals and knot spaces

“…One could ask whether there is another multiplication on configuration space and one on our total space (both compatible with each other and with connect-sum) which extends to a little 2-cubes action compatible with Budney's 2-cubes action on K. Assuming any such 2-cubes action on these spaces, an argument analogous to the proof of Theorem 2 shows that the evaluation of a Bott-Taubes type class on the bracket of any two classes is zero, using the fact that C q D C q .R 3 / has cohomology only in even dimensions. (So this argument only extends to R n for odd n.) In view of Sakai's result in [21], this strongly suggests that no such lift of the 2-cubes action exists.…”

“…The author would like to express deep gratitude to Ralph Cohen whose ideas, enthusiasm and advice were indispensable for the completion of this article. He would also like to thank Nathan Habegger, Pascal Lambrechts, Paolo Salvatore, Dev Sinha and Ismar Volić for enlightening conversations relating to the subject matter of this paper, and the referee for informing him about Sakai's result in [21].…”

An integral expression of the first nontrivial one-cocycle of the space of long knots in ℝ3

“…It is shown in [Cattaneo et al 2002] that, when n > 3, the induced map I on cohomology restricted to the space of trivalent graph cocycles is injective. In [Sakai 2008], the author gave the first example of a nontrivalent graph cocycle ( Figure 1) which also gives a nonzero class…”

“…The support of (antisymmetric) vol S 2 is contained in a sufficiently small neighborhood of the poles (0, 0, ±1) as in [Sakai 2008]. So only the configurations with the images of the Gauss maps lying in a neighborhood of (0, 0, ±1) can nontrivially contribute to various integrals below.…”

“…The cocycles of n corresponding to trivalent graph cocycles via I generalize an integral expression of finite type invariants for (long) knots in ‫ޒ‬ 3 [Altschuler and Freidel 1997;Bott and Taubes 1994;Kohno 1994;Volić 2007]. In [Sakai 2008] the author found a nontrivalent graph cocycle ∈ Ᏸ * and proved that, when n > 3 is odd, it gives a nonzero cohomology class [I ( )] ∈ H 3n−8 D R ( n ). On the other hand, when n = 3, some obstructions to I being a cochain map (called anomalous obstructions; see for example [Volić 2007, Section 4.6]) may survive, so even the closedness of I ( ) was not clear.…”