TITLE: 3-manifold with fundamental group $\mathbb Z$
QUESTION [5 upvotes]: Let $M$ be a compact $3$-manifold with nonempty boundary. If $\pi_1(M)=\mathbb Z$, can we prove that $M$ is homeomorphic to $S^1 \times D^2$?

REPLY [19 votes]: No.  For example, take a copy of $S^1 \times S^2$ and remove the interior of a closed, nicely embedded, three-ball.
You will need to add the hypothesis of irreducibility (to rule out "punctures" as in my example immediately above) and the hypothesis of orientability (to rule out the solid Klein bottle).  These hypotheses, plus the disk theorem, gives the desired result.
See Proposition 3.4 of Hatcher's three-manifold notes, Exercise 5.3 in Hempel's book, or Exercise I.32 in Jaco's book.