Varieties of chord diagrams, braid group cohomology and degeneration of equality conditions

Victor A. Vassiliev

Vol. 326 (2023), No. 1, 135–160

DOI: 10.2140/pjm.2023.326.135

Abstract

For any finite-dimensional vector space $\mathcal{ℱ}$ of continuous functions $f:{ℝ}^1\to {ℝ}^1$ we consider subspaces in $\mathcal{ℱ}$ defined by systems of equality conditions $f(a_i)=f(b_i)$, where $\{a_i,b_i\}$, $i=1,\dots <!--FUNCTION\; APPLICATION-->,n$, are some pairs of points in ${ℝ}^1$. It is proven that if $\dim <!--FUNCTION\; APPLICATION-->\mathcal{ℱ}<2n-I(n)$, where $I(n)$ is the number of ones in the binary notation of $n$, then there necessarily exist 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 some of these subspaces.

Next, we apply these estimates to knot theory (in which systems of equality conditions are known as chord diagrams) and prove the inevitable presence of complicated nonstable terms in sequences of spectral sequences computing cohomology groups of spaces of knots.

Keywords

chord diagram, configuration space, characteristic class

Mathematical Subject Classification

Primary: 55R80

Secondary: 57R22

Milestones

Received: 26 January 2023

Revised: 6 July 2023

Accepted: 21 October 2023

Published: 14 December 2023