弦图的种类、辫群上同调与等式条件的退化
Varieties of chord diagrams, braid group cohomology and degeneration of equality conditions
AI总结：
该研究证明当连续函数空间维数满足特定条件时，等式条件系统定义的子空间余维数会超出条件数，给出余维下降下界，并将其应用于纽结理论，证实纽结空间上同调谱序列中必然存在复杂非稳定项。
摘要：
对于任意由连续函数$f:{\n\bbr}^1 \to {\n\bbr}^1$构成的有限维向量空间${\n\bcal F}$，考虑由等式条件系统$f(a_i) = f(b_i)$在${\n\bcal F}$中定义的子空间，其中$(a_i, b_i)$（$i=1, \n\tdots, n$）是${\n\bbr}^1$中的若干点对。研究证明：若$\n\bdim {\n\bcal F} < 2n-I(n)$（其中$I(n)$是$n$的二进制表示中1的个数），则必然存在由$n$个等式条件构成的独立系统，其在${\n\bcal F}$中定义的子空间余维数大于$n$。我们还证明了这些子空间余维数必然下降幅度的下界估计。\n 这些估计随后被应用于纽结理论（在该领域中，等式条件系统被称为弦图（chord diagram））。研究证明，在计算纽结空间上同调群的谱序列序列中，复杂非稳定项的存在是必然的。
英文摘要：
For any finite-dimensional vector space ${\mathcal F}$ of continuous functions $f:{\mathbb R}^1 \to {\mathbb R}^1$ consider subspaces in ${\mathcal F}$ defined by systems of {\em equality conditions} $f(a_i) = f(b_i)$, where $(a_i, b_i)$, $i=1, \dots, n$, are some pairs of points in ${\mathbb R}^1$. It is proved that if $\dim {\mathcal F} < 2n-I(n)$, where $I(n)$ is the number of ones in the binary notation of $n$, then there necessarily are independent systems of $n$ equality conditions defining the subspaces of codimension greater than $n$ in ${\mathcal F}$. We also prove lower estimates of the sizes of the inevitable drops of the codimensions of these subspaces. These estimates are then applied to knot theory (in which systems of equality conditions are known as {\em chord diagrams}). The inevitable presence of complicated non-stable terms in sequences of spectral sequences calculating the cohomology groups of spaces of knots is proved. Keywords: Chord diagram, configuration space, characteristic class
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