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

IF 0.6 4区 数学 Q3 MATHEMATICS
Syunji Moriya
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引用次数: 0

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.
辛哈的一维长节谱序列和小二盘运算符的非形式性
我们计算了辛哈在标度为一的长结空间模化浸入的同调谱序列的一些微分,主要是在特征为 2 或 3 的域上。这个谱序列与瓦西里耶夫的标度为 $\geq2$ 的长节空间谱序列密切相关。我们证明了一个元素的 d2 微分在特征 2 中不为零,这一点萨尔瓦托雷已经基本证明了,而另一个元素的 d3 微分在特征 3 中不为零。虽然在标度一中序列的几何意义并不明确,但这些结果在操作数的非形式性方面有一些应用。特征 3 中的结果意味着标准映射 $C_\ast(E_1)\to C_\ast(E_2)$ 在特征 3 中的平面非形式性,其中 $C_\ast(E_k)$ 表示链小 k 盘操作数。我们还利用特征 2 的结果重新证明了萨尔瓦托雷的结果,即在特征 2 中$C_\ast(E_2)$ 不是形式的平面操作数。为了进行计算,我们通过配置空间和胖对角线之间的对偶性,把谱序列背后的配置空间结构转移到胖对角线上的托姆空间上。这一过程使我们能够通过相对简单的映射来描述托姆空间的微分。我们还证明,在特征 $\not=2$ 中,双阶 $(-4,2)$ 的生成器的 d2 微分为零。这一计算说明了如何利用描述来处理三项关系。虽然本文的计算集中于一维,但我们的方法同样适用于一维$\geq2$,而且我们为一般维度准备了大部分基本概念和定理。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
36
审稿时长
6-12 weeks
期刊介绍: The Quarterly Journal of Mathematics publishes original contributions to pure mathematics. All major areas of pure mathematics are represented on the editorial board.
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