Talk:Three dimensional Heisenberg group (Ex)

Let $T$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}T$ be a torus and let $y$$y$ and $z$$z$ be generators of $\pi_1(T)\cong\mathbb{Z}\oplus\mathbb{Z}$$\pi_1(T)\cong\mathbb{Z}\oplus\mathbb{Z}$. Let $f:T\rightarrow T$$f:T\rightarrow T$ be a homeomorphism of the torus such that $f_*(y)=yz^{-1}$$f_*(y)=yz^{-1}$ and $f_*(z)=z$$f_*(z)=z$ (ie. $f$$f$ is a Dehn twist.) Let $M$$M$ be a torus bundle over $S^1$$S^1$ with monodromy $f$$f$. Then $\pi_1(M)$$\pi_1(M)$ is an HNN extension of $\pi_1(T)$$\pi_1(T)$ and we can write down a presentation $\pi_1(M) = \langle x,y,z \mid x^{-1}yx=yz^{-1}, x^{-1}zx=z, [y,z]=1\rangle$$\pi_1(M) = \langle x,y,z \mid x^{-1}yx=yz^{-1}, x^{-1}zx=z, [y,z]=1\rangle$, with stable letter $x$$x$. This is equivalent to the given presentation of the Heisenberg group. Therefore $M$$M$ is the desired closed, orientable $3$$3$-manifold.