TITLE: Degree 1 maps from $S^1 \times S^2$
QUESTION [5 upvotes]: Suppose that $M$ is a closed connected 3-manifold and there is a degree 1 map $f: S^1 \times S^2 \to M$.  Does it follow that $M \cong S^1 \times S^2$?  I know that $\pi_1 M$ is cyclic since $f$ must be surjective on $\pi_1$.

REPLY [4 votes]: An addendum to Goa'uld's answer: If $M$ is a closed oriented connected 3-manifold which admits a degree 1 map $S^2\times S^1\to M$, then either $M$ is $S^3$ or $M=S^2\times S^1$. See this paper where it is proven that nontrivial lens spaces cannot be targets of degree 1 maps from $S^2\times S^1$.