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On the cohomology of varieties of chord diagrams

V. A. Vassiliev

TL;DR

This work determines the mod $2$ cohomology ring of the closure $\overline{CD_2}$ of the space of codimension-two subalgebras of $C^ ∞(S^1, \mathbb{R})$ defined by chord diagrams and computes the Stiefel--Whitney classes of its canonical normal bundle. Through a detailed CW-decomposition and multiple geometric realizations of its homology (including a Klein bottle fibration), it establishes nontrivial $w_1$ and $w_2$ with the total class $w = 1+W+W^2$, and derives Borsuk--Ulam-type consequences for linear combinations of smooth functions. The results give explicit obstructions to trivializing the bundle over key cycles and connect to interpolation and knot-theoretic contexts via the algebraic structure of chord diagrams. Overall, the paper provides a precise topological description of $\overline{CD_2}$ and concrete Stiefel--Whitney obstructions with potential applications in related geometric problems.

Abstract

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

On the cohomology of varieties of chord diagrams

TL;DR

This work determines the mod cohomology ring of the closure of the space of codimension-two subalgebras of defined by chord diagrams and computes the Stiefel--Whitney classes of its canonical normal bundle. Through a detailed CW-decomposition and multiple geometric realizations of its homology (including a Klein bottle fibration), it establishes nontrivial and with the total class , and derives Borsuk--Ulam-type consequences for linear combinations of smooth functions. The results give explicit obstructions to trivializing the bundle over key cycles and connect to interpolation and knot-theoretic contexts via the algebraic structure of chord diagrams. Overall, the paper provides a precise topological description of and concrete Stiefel--Whitney obstructions with potential applications in related geometric problems.

Abstract

We study the space of codimension two subalgebras in defined by pairs of conditions , , 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.

Paper Structure

This paper contains 6 sections, 13 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Cell decomposition and homology group of $\overline{CD_2}$
  3. Other realizations of homology groups
  4. Proof of Theorem \ref{['SWth']} and Propositions \ref{['nz']} and \ref{['BU']}.
  5. Proof of Theorem \ref{['mthmmult']}
  6. Proof of Proposition \ref{['corm']}

Key Result

Theorem 1

The mod 2 cohomology groups of the space $\overline{CD_2}$ are isomorphic to ${\mathbb Z}_2$ in dimensions 0, 1, 2, and 3, and are trivial in all other dimensions.

Theorems & Definitions (17)

  • Definition 1
  • Example 1
  • Example 2
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Corollary 2
  • Proposition 1
  • Proposition 2
  • ...and 7 more