On torsion in linearized Legendrian contact homology

TL;DR

This paper constructs examples of Legendrian submanifolds in standard contact space with prescribed torsion in their linearized Legendrian contact (co)homology groups over integers, demonstrating the presence of algebraic torsion.

Contribution

It provides explicit constructions of Legendrian submanifolds with any finitely generated abelian group as torsion in their linearized contact homology, for dimensions n ≥ 3, n ≠ 4.

Findings

Existence of Legendrian submanifolds with prescribed torsion in contact homology.

Explicit examples for any finitely generated abelian group G.

Non-vanishing algebraic torsion in linearized Legendrian contact (co)homology.

Abstract

In this short note we discuss certain examples of Legendrian submanifolds, whose linearized Legendrian contact (co)homology groups over integers have non-vanishing algebraic torsion. More precisely, for a given arbitrary finitely generated abelian group $G$ and a positive integer $n\geq 3$, $n\neq 4$, we construct examples of Legendrian submanifolds of the standard contact vector space $\mathbb R^{2n+1}$, whose $n-1$-th linearized Legendrian contact (co)homology over $\mathbb Z$ computed with respect to a certain augmentation is isomorphic to $G$.