Sinha's spectral sequence for long knots in codimension one and non-formality of the little 2-disks operad

Abstract

We compute some differentials of Sinha's spectral sequence for cohomology of the space of long knots modulo immersions in codimension one, mainly over a field of characteristic $2$ or $3$. This spectral sequence is closely related to Vassiliev's spectral sequence for the space of long knots in codimension $\geq 2$. We prove that the $d_2$-differential of an element is non-zero in characteristic $2$, which has already essentially been proved by Salvatore, and the $d_3$-differential of another element is non-zero in characteristic $3$. While the geometric meaning of the sequence is unclear in condimension one, these results have some applications to non-formality of operads. The result in characteristic $3$ implies planar non-formality of the standard map $C_*(E_1)\to C_*(E_2)$ in characteristic $3$, where $C_*(E_k)$ denotes the chain little $k$-disks operad. We also reprove the result of Salvatore which states that $C_*(E_2)$ is not formal as a planar operad in characteristic $2$ using the result in characteristic $2$. For the computation, we transfer the structure on configuration spaces behind the spectral sequence onto Thom spaces over fat diagonals through a duality between configuration spaces and fat diagonals. This procedure enables us to describe the differentials by relatively simple maps to Thom spaces. We also show that the $d_2$-differential of the generator of bidegree $(-4,2)$ is zero in characteristic $\not= 2$. This computation illustrates how one can manage the 3-term relation using the description. Although the computations in this paper are concentrated to codimension one, our method also works for codimension $\geq 2$ and we prepare most of basic notions and lemmas for general codimension.