A class of incompressible surfaces in 3-manifolds fibering over \(S^ 1\) (Q1191668)

Main Theorem. Let \(M=F\times_ \varphi S^ 1\), \(g(F)\geq 2\), where \(F\) is an orientable closed surface, \(\varphi\) preserves the orientation of \(F\), and \(\text{rank}(H_ 1(M,\mathbb{Z}))\geq 2\). Then there are two simple closed curves \(\ell_ 1\), \(\ell_ 2\) on \(F\times\partial I\), \(\ell_ 1\to F\times\{0\}\), \(\ell_ 2\to F\times\{1\}\), such that \(\varphi(\ell_ 1)=\ell_ 2\). For any integer \(N\geq 0\), there exists a surface \(G\hookrightarrow F\times I\), such that \(g(G)\geq N\), \(\partial G=\ell_ 1\cup\ell_ 2\), \(p|_ G\), the projection \(p: F\times I\to F\) restricted on \(G\), is an immersion, \(M\) is fibered over \(S^ 1\) with a fiber \(G/\varphi\). Moreover, \(G/\varphi\hookrightarrow F\times I/\varphi\) is an incompressible surface.