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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!!

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    $\begingroup$ What about the complex projective plane minus a ball? That is homotopic to a 2-sphere and the boundary is a 3-sphere. $\endgroup$
    – Will Sawin
    Commented Dec 7, 2021 at 0:59
  • $\begingroup$ I have edited the question. I think the edited question is still silly. S^2 \times D^1 \times D^1 will work. $\endgroup$
    – piper1967
    Commented Dec 7, 2021 at 1:16
  • $\begingroup$ Maybe use the homology long exact sequence of the pair $(M,\partial M)$. $\endgroup$ Commented Dec 7, 2021 at 1:41
  • $\begingroup$ Given two of those, you can glue a 3-ball on the boundary of one to a 3-ball on the boundary of the other. Iterating this, you can make $b_1$ of the boundary into any natural number. $\endgroup$
    – Will Sawin
    Commented Dec 7, 2021 at 2:55

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