Talk:Three dimensional Heisenberg group (Ex)

Revision as of 12:48, 7 January 2019

Let $<wikitex>; Let $T$ be a torus and let $y$ and $z$ be generators of $\pi_1(T)\cong\mathbb{Z}\oplus\mathbb{Z}$. Let $f:T\rightarrow T$ be a homeomorphism of the torus such that $f_*(y)=yz^{-1}$ and $f_*(z)=z$ (ie. $f$ is a Dehn twist.) Let $M$ be a torus bundle over $S^1$ with monodromy $f$. Then $\pi_1(M)$ is an HNN extension of $\pi_1(T)$ and we can write down a presentation $\pi_1(M) = <x,y,z | x^{-1}yx=yz^{-1}, x^{-1}zx=z, [y,z]=1>$, with stable letter $x$. This is equivalent to the given presentation of the Heisenberg group. Therefore $M$ is the desired closed, orientable $-manifold. </wikitex> [[Category:Exercises]] [[Category:Exercises with solution]]T$ be a torus and let $y$ and $z$ be generators of $\pi_1(T)\cong\mathbb{Z}\oplus\mathbb{Z}$. Let $f:T\rightarrow T$ be a homeomorphism of the torus such that $f_*(y)=yz^{-1}$ and $f_*(z)=z$ (ie. $f$ is a Dehn twist.) Let $M$ be a torus bundle over $S^1$ with monodromy $f$. Then $\pi_1(M)$ is an HNN extension of $\pi_1(T)$ and we can write down a presentation $\pi_1(M) = \langle x,y,z \mid x^{-1}yx=yz^{-1}, x^{-1}zx=z, [y,z]=1\rangle$, with stable letter $x$. This is equivalent to the given presentation of the Heisenberg group. Therefore $M$ is the desired closed, orientable $3$-manifold.