Let $X$ be a smooth 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 $\pi\_1 \ne 0$? 



$\pi_1$ of 4-manifolds that "look like" disk bundles
Let's take your $p$ to be prime. Then $X$ has to be simply connected, even without all of the hypotheses. Here is the argument. 


From the map on $\pi\_1(L) \to Z\_p$ you get a map $L \to BZ\_p$. This map is clearly $0$ in $H\_3$, since it factors through the inclusion of $L$ into the $4$-manifold $X$. On the other hand, it's well-known that $L$ generates $H\_3(BZ\_p) \cong Z\_p$. (A quick explanation: you can build $BZ\_p$ by attaching a 4-cell by a map of degree $p$ (the universal covering $S^3 \to L$) and then higher cells.)


I think a similar proof works if $p=mn$ with $m$ and $n$ relatively prime, and $\pi\_1(X) = Z\_n$. You have to know a little more about the map $H\_3(BZ\_p) \to H\_3(BZ\_n)$ induced by a surjection. I'm pretty sure it's given by multiplication by $m^2$.


