On the cohomology of varieties of chord diagrams

We study the space of codimension two subalgebras in $C^{\mathrm{\infty }}(S^1,\mathbb{R})$ defined by pairs of conditions $f(\phi )=f(\psi )$, $\phi \mathrm{\ne }\psi \in S^1$, or by their limits. We compute the mod 2 cohomology ring of this space, and also the Stiefel–Whitney classes of the tautological vector bundle on it.