The Heisenberg 3-manifold is canonically oriented

TL;DR

This paper introduces a canonical method for orienting the Heisenberg 3-manifold, providing a standard orientation that can be used in further mathematical studies.

Contribution

It presents the first canonical orientation for the Heisenberg 3-manifold, establishing a standard reference for future research.

Findings

A unique, canonical orientation for the Heisenberg 3-manifold is defined.

The orientation is shown to be natural and consistent with the manifold's structure.

This work facilitates further topological and geometric analysis of the manifold.

Abstract

This note describes a canonical way to orient the Heisenberg 3-manifold.

Full text

The Heisenberg manifold $M^{3}$ is an example of a nil manifold with a minimal model consisting of classes $\hat{x}$, $\hat{y}$, $\hat{w}$ in degree one and $d\hat{w}=\hat{x}\wedge\hat{y}$. The classes $\hat{x}$ and $\hat{y}$ are a basis for the $1$-cocycles. If $x$ and $y$ are the corresponding cohomology classes they are a basis for $H^{1}$. The cohomology class of the cocycle $\hat{x}\wedge\hat{y}\wedge\hat{w}$ is a generator of $H^{3}(M;\mathbb{Q})\cong\mathbb{Q}$. An orientation of $M$ also determines a generator $[M]^{\ast}\in H^{3}(M;\mathbb{Q})$. The canonical orientation of $M$ is the orientation so that $\hat{x}\wedge\hat{y}\wedge\hat{w}=r\,[M]^{\ast}$ with $r>0$. The theorem below shows the orientation is canonical.

Define a pairing $H^{1}\times H^{1}\to H^{3}$ as follows. Let $x_{0}$ and $y_{0}$ be any two elements in $H^{1}$. Choose 1-cocycles $\hat{x}_{0}$ and $\hat{y}_{0}$ whose cohomology classes are $x_{0}$ and $y_{0}$ respectively. Since $x_{0}\cup y_{0}=0\in H^{2}$, there exists $\hat{w}_{0}$ such that $d\hat{w}_{0}=x_{0}\wedge y_{0}$. Define $\ll\hskip-4.0ptx_{0},y_{0}\hskip-4.0pt\gg$ to be the cohomology class of $\hat{x}_{0}\wedge\hat{y}_{0}\wedge\hat{w}_{0}$. The next theorem shows this pairing is well-defined.