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

TL;DR

This paper investigates the structure of subspaces defined by equality conditions in function spaces, establishing dimension bounds and applying these results to knot theory and cohomology of knot spaces.

Contribution

It introduces new bounds on the codimensions of subspaces defined by equality conditions and applies these to analyze the stability of spectral sequences in knot cohomology.

Findings

Existence of systems with codimension greater than n under certain dimension constraints

Lower bounds on the size of codimension drops in subspace sequences

Presence of non-stable terms in spectral sequences for knot cohomology

Abstract

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