Nonorientable, incompressible surfaces of genus 3 in \(M_{\phi}(\lambda /\mu)\) manifolds (Q761664)

Let \(M_{\phi}\) be a once-punctured torus bundle over \(S^ 1\) with a hyperbolic monodromy map \(\phi\in SL(2,Z):\) that is \(M_{\phi}=F\times R/(x,t)\sim (\phi (x),t+1)\) where F denotes a punctured torus. Let \(\phi\) : \(F\times R\to M_{\phi}\) be the natural projection. Since \(\partial M_{\phi}\) is a torus \(H_ 1(\partial M_{\phi})\) is generated by two elements m,\(\ell\) as follows: The second generator \(\ell\) of \(H_ 1(\partial M_{\phi})\) is determined by the boundary curve of a fiber F (with the clockwise orientation). If tr \(\phi\) \(>0\), then the restriction of \(\phi\) to the boundary of a fiber has a fixed point \(\alpha\). Then the first generator m of \(H_ 1(\partial M_{\phi})\) is determined by the circle \(\phi\) (L), where L is the straight line in \(\partial F\times R\) which joins (\(\alpha\),0) and (\(\alpha\),1). If tr \(\phi\) \(>0\), then the restriction of -\(\phi\) to \(\partial F=\{e^{i\theta}|\) \(0\leq \theta <2\pi \}\), has a fixed point \(\alpha\), say \(\alpha =e^{ia}\). Let L be the curve on \(\partial F\times R\) given by the equation \(z=e^{i(t+a)}\) where \(z\in \partial F\) and \(t\in R\). Then L joins (\(\alpha\),0) and (- \(\alpha\),1) with a negative half twist with respect to the chosen orientation of \(\partial F\). The first generator m is determined by the curve \(\phi\) (L). If a curve on \(\partial M_{\phi}\) represents a homology class \(am+b\ell\) in \(H_ 1(\partial M_{\phi})\), then the slope of the curve is given by b/a. \(M_{\phi}(\lambda /\mu)\) is the manifold obtained from \(M_{\phi}\) by Dehn surgery along a curve on \(\partial M_{\phi}\) of slope \(\lambda\) /\(\mu\). The author shows the following Theorem: Let \(M_{\phi}(\lambda /\mu)\) be an irreducible, non- Haken, orientable 3-manifold. Let \(K_ 1\) and \(K_ 2\) be nonorientable, incompressible, genus 3 surfaces in \(M_{\phi}(\lambda /\mu)\) such that the classes of \(K_ 1\) and \(K_ 2\) are equal in \(H_ 2(M_{\phi}(\lambda /\mu),Z_ 2^{\mu})\). Then \(K_ 1\) is isotopic to \(K_ 2\) with the possible exception of the case of \(M_{\phi}(1/1)\) with \(\phi =-{\bar \alpha}^ 2\beta {\bar \alpha}^ 2\beta {\bar \alpha}^ 2\beta\), where \(\alpha =\left( \begin{matrix} 1\quad 1\\ 0\quad 1\end{matrix} \right)\) \(\beta =\left( \begin{matrix} 1\quad 0\\ 1\quad 1\end{matrix} \right)\). We identify \(\phi\) with its equivalence class defined by conjugation and taking the inverse.