Nonexistence of Bonatti-Langevin examples of Anosov flows on closed four manifolds (Q864446)

A submanifold \(S\) of \(M\) is incompressible if its fundamental group injects into the one of \(M\). Examples of incompressible submanifolds are the closed submanifolds transverse to Anosov flows on closed 3-manifolds, and the global sections for flows. The authors prove the following theorem: There are no Anosov flows on closed 4-manifolds exhibiting a closed, incompressible, transverse submanifold intersecting all orbits except finitely many closed ones. The proof presented in the paper works only for manifolds of dimension four. It is based on studying of the trace of the weak stable foliation of the flow on the transverse submanifold.