Boundary of a $4$-manifold and the fundamental group I am trying to learn $4$-manifolds with boundaries and I don't know much about this topic so these questions may be silly. Given a $4$-manifold $M$ with a boundary say $N$,

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*Assume $\pi_1(N)$ is known, is there a way to compute $\pi_1(M)$ or is it possible to say any features of $\pi_1(M)$? To be more precise how the boundary is influencing the fundamental group of the manifold?


*Can I put some restriction on $N$ so that $\pi_1(M)$ becomes finite?
I apologize for asking too many questions in a single post. Any techniques, references, or suggestions will be helpful.
Thanks in advance!
 A: Let me expand the comment of @RyanBudney by giving concrete examples.
Given coprime positive integers $p,q$ and $r$, Brieskorn spheres are important classes of $3$-manifolds, they can be defined as the links of singularities at the origin: $$\Sigma(p,q,r) = \{ x^p +y^q +z^r =0 \} \cap S^5 \subset \mathbb C^3.$$
The fundamental groups of Brieskorn spheres are well-known, see for instance the paper of Milnor.
However, the following question is a very central problem in low-dimensional topology, see Kirby's problem list, Problem 4.20:

Which homology 3-spheres (in particular Brieskon spheres) bound smooth contractible 4-manifolds?

In some cases, Brieskorn spheres may bound Mazur type contractible manifolds which can be built by a single $0$-, $1$-, and $2$-handle.
Once you know the Kirby diagram of a Mazur manifold, you can both compute the fundamental groups of the $4$-manifold and its boundary $3$-manifold, see the dicussion in this question.
A: For $2.$, there is no condition purely on the topology of $N$ which makes $\pi_{1}(M)$ finite.
Take any $4$-manifold $M$ with boundary $N$. Then taking a point not on the boundary and make a connected sum with $T^4$ makes $\pi_{1}$ infinite.
There may be some condition which is in terms of the topology of $N$ and properties of the inclusion $i: N \rightarrow M$. But taking the question literally as stated the answer to $2.$ is no.
