$3$-manifolds fibering over $S\sp{1}$

Abstract

Let M be a closed 3-manifold that is the total space of a fiber bundle with base S1 and fiber the closed 2-manifold F. Assume that genus (F) > 2 if F is orientable, and that genus (F) > 3 if F is nonorienta- ble. We say that M has unique fiber over S1 if, for any fibering of M over S1 with fiber F', we have F' -_ F. We prove that M has unique fiber over S1 if and only if rank (HI (M; Z)) = 1. In the case that rank (HI (M; Z)) # 1, M fibers over S1 with fiber any of infinitely many distinct closed surfaces. In (5), Tollefson proved that if M is a 3-manifold of the form F x S1, where F is a closed oriented surface of genus g > 2, then M fibers over Sl with fiber any of infinitely many distinct surfaces. We extend this result to a characteri- zation of uniqueness of the fiber in 3-manifolds fibering over Sl. All manifolds and maps considered will be differentiable (say C1). We say that the closed 3-manifold Mfibers over Sl with fiber F if M is diffeomorphic to the quotient manifold (F x I)/d obtained by identifying F x {O} Q F