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 $\ge 2$. 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 $C_{\ast }(E_1)\to C_{\ast }(E_2)$ 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 $d_2$-differential of the generator of bidegree $(-4,2)$ is zero in characteristic $\ne 2$. 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 $\ge 2$ and we prepare most of basic notions and lemmas for general codimension.

40 pages, 11 figures, Formerly named "Differentials of Sinha's spectral sequence for long knots in codimension one". Corrected errors in the published version, see subsection 1.1. Main results are still valid