[Paper Review] On the pure virtual braid group $PV_3$

В. Г. Бардаков, Roman Mikhailov+2 more|ArXiv.org|Jun 9, 2009

Geometric and Algebraic Topology15 references4 citations

TL;DR

This paper investigates the pure virtual braid group $PV_3$, establishing its free product decomposition, residual torsion-free nilpotence, asphericity, cohomology ring structure, and associated graded Lie algebra. The key contribution is proving that $PV_3$ is residually torsion-free nilpotent, implying completeness of finite type invariants for 3-strand virtual pure braids.

ABSTRACT

In this article, we investigate various properties of the pure virtual braid group PV_3. From its canonical presentation, we obtain a free product decomposition of PV_3. As a consequence, we show that PV_3 is residually torsion free nilpotent, which implies that the set of finite type invariants in the sense of Goussarov-Polyak-Viro is complete for virtual pure braids with three strands. Moreover we prove that the presentation of PV_3 is aspherical. Finally we determine the cohomology ring and the associated graded Lie algebra of PV_3.

Motivation & Objective

• To analyze the algebraic and topological structure of the pure virtual braid group $PV_3$.
• To determine whether $PV_3$ is residually torsion-free nilpotent, which would imply completeness of finite type invariants for virtual pure braids.
• To establish the asphericity of the presentation complex of $PV_3$.
• To compute the integral cohomology ring and the associated graded Lie algebra of $PV_3$.

Proposed method

• Derive a free product decomposition of $PV_3$ from its canonical presentation.
• Use the decomposition to prove that $PV_3$ is residually torsion-free nilpotent via induction and properties of Malcev completions.
• Construct a presentation complex for $PV_3$ and prove it is aspherical using the presentation's geometric and combinatorial properties.
• Compute the cohomology ring $H^*(PV_3; Z)$ as a quotient of an exterior algebra with specific relations derived from the group's relations.
• Determine the associated graded Lie algebra $L(PV_3)$ as a quotient of a free Lie algebra modulo relations from the group's lower central series.
• Apply the 5-Lemma to a commutative diagram of Lie algebras to verify the structure of $L(PV_3)$.

Experimental results

Research questions

• RQ1 Is the pure virtual braid group $PV_3$ residually torsion-free nilpotent?
• RQ2 Is the standard presentation of $PV_3$ aspherical?
• RQ3 What is the structure of the integral cohomology ring $H^*(PV_3; Z)$?
• RQ4 What is the associated graded Lie algebra of $PV_3$?
• RQ5 How do the relations in $PV_3$ constrain the cohomology and Lie algebra structures?

Key findings

• The group $PV_3$ admits a free product decomposition, which implies it is residually torsion-free nilpotent.
• $PV_3$ is aspherical, meaning its presentation complex has trivial higher homotopy groups.
• The cohomology ring $H^*(PV_3; Z)$ is isomorphic to the exterior algebra on six generators modulo three specific relations: $ abla_{ij}^* abla_{ji}^* = 0$, $( abla_{13}^* - abla_{31}^*) abla_{ij}^* = ( abla_{12}^* - abla_{21}^*) abla_{ij}^* + ( abla_{23}^* - abla_{32}^*) abla_{ij}^*$, and $ abla_{21}^* abla_{31}^* eq abla_{21}^* abla_{32}^* + abla_{23}^* abla_{31}^*$.
• The associated graded Lie algebra $L(PV_3)$ is the quotient of the free Lie algebra on five generators $A_1, B_1, A_2, B_2, C_1$ modulo the relations $[A_1,B_1] = 0$, $[A_2,B_2] = 0$, and $[C_1, X] = [Y, X]$ for specific $X, Y$ corresponding to group relations.
• The cohomology ring relations are linearly dependent, with five relations collapsing to two independent ones in the quotient.
• The structure of $L(PV_3)$ is fully determined by the group's lower central series and the action of the central element $c_1$ on the free part.