Talk:Three dimensional Heisenberg group (Ex)
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Josh Howie (Talk | contribs) (Created page with "<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 ...") |
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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 | 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) = | + | $\pi_1(M) = \langle x,y,z \mid x^{-1}yx=yz^{-1}, x^{-1}zx=z, [y,z]=1\rangle$, with stable letter $x$. |
This is equivalent to the given presentation of the Heisenberg group. Therefore $M$ is the desired closed, orientable $3$-manifold. | This is equivalent to the given presentation of the Heisenberg group. Therefore $M$ is the desired closed, orientable $3$-manifold. | ||
</wikitex> | </wikitex> | ||
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Latest revision as of 12:52, 7 January 2019
Let be a torus and let and be generators of . Let be a homeomorphism of the torus such that and (ie. is a Dehn twist.) Let be a torus bundle over with monodromy . Then is an HNN extension of and we can write down a presentation , with stable letter . This is equivalent to the given presentation of the Heisenberg group. Therefore is the desired closed, orientable -manifold.