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

Abstract

Given a finite-dimensional space \({\mathcal F}\) of continuous functions \({\mathbb R}^1 \to {\mathbb R}^1\), we study the subspaces in \({\mathcal F}\) defined by systems of \(n\) equality conditions \(f(a_i) = f(b_i)\) for varying collections of pairs of points \(a_i, b_i \in {\mathbb R}^1\). If \({\mathcal F}\) is generic and its dimension is large enough compared with \(n\) then any \(n\) independent conditions of this type define a subspace of codimension exactly \(n\). On contrary, we prove that this property necessarily fails if \(\dim {\mathcal F} < 2n-I(n)\) where \(I(n)\) is the number of ones in the binary notation of \(n\). In particular if \(n\) is a power of 2 then the minimal dimension of \({\mathcal F}\) for which this property can hold is equal to \(2n-1\). Keywords: Chord diagram, configuration space, characteristic class