Let $X$ be a compact oriented 4-manifold with $\partial X=L(p,1)$, $H_2(X,\Bbb Z)=\Bbb Z$, $H_3(X, \Bbb Z)=0$ and the induced map $\pi_1(L(p,1)) \to X$ surjective. What are the possibilities for $\pi_1(X)$? In particular, are there examples where $\pi_1 \ne 0$?