On the cohomology of varieties of chord diagrams

TL;DR

This paper investigates the cohomology and characteristic classes of a space of codimension two subalgebras in smooth functions on the circle, defined by specific pointwise conditions.

Contribution

It computes the mod 2 cohomology ring and Stiefel--Whitney classes of the tautological bundle for this particular space of subalgebras.

Findings

Computed the mod 2 cohomology ring of the space.

Determined the Stiefel--Whitney classes of the tautological bundle.

Provided explicit descriptions of the cohomological invariants.

Abstract

We study the space of codimension two subalgebras in $C^\infty(S^1, {\mathbb R})$ defined by pairs of conditions $f(\varphi)=f(\psi)$, $\varphi \neq \psi \in S^1$, 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.