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 $\geq2$. We prove that the d2-differential of an element is non-zero in characteristic 2, which has already essentially been proved by Salvatore, and the d3-differential of another element is non-zero in characteristic 3. While the geometric meaning of the sequence is unclear in codimension one, these results have some applications to non-formality of operads. The result in characteristic 3 implies planar non-formality of the standard map $C_\ast(E_1)\to C_\ast(E_2)$ in characteristic 3, where $C_\ast(E_k)$ denotes the chain little k-disks operad. We also reprove the result of Salvatore which states that $C_\ast(E_2)$ 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 d2-differential of the generator of bidegree $(-4,2)$ is zero in characteristic $\not=2$. This computation illustrates how one can manage the three-term relation using the description. Although the computations in this paper are concentrated to codimension one, our method also works for codimension $\geq2$ and we prepare most of the basic notions and lemmas for general codimension.