Let $M$ be a compact $4-$manifold with boundary $dM$. If $M$ has the homotopy type of a wedge of $2-$spheres then is it always true that $b_1(dM)=0$? By $b_1$, I mean first Betti number.
It is known that any closed, oriented $3-$manifold is the boundary of a compact $4-$manifold. However, any homological (singular homology with $\mathbb{Q}$ or $\mathbb{Z}$-coefficient) relation between boundary and the parent manifold I don't know.
Thanks in advance!!