  We give two infinite families of examples of closed, orientable, irreducible
3-manifolds $M$ such that $b_1(M)=1$ and $\pi_1(M)$ has weight 1, but $M$ is
not the result of Dehn surgery along a knot in the 3-sphere. This answers a
question of Aschenbrenner, Friedl and Wilton, and provides the first examples
of irreducible manifolds with $b_1=1$ that are known not to be surgery on a
knot in the 3-sphere. One family consists of Seifert fibered 3-manifolds, while
each member of the other family is not even homology cobordant to any Seifert
fibered 3-manifold. None of our examples are homology cobordant to any manifold
obtained by Dehn surgery along a knot in the 3-sphere.
