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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 $N$,

  1. 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?

  2. 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!

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    $\begingroup$ Generally there is no relation between the two fundamental groups. But if you have a handle presentation of $M$, i.e. building $M$ by attaching handles to $N \times [0,1]$ then you will have relations. The simpler the handle presentation, the simpler the relation. This gives you a very weak answer to (2), as well. $\endgroup$ May 27, 2022 at 0:41
  • $\begingroup$ About 2: If you know that $M$ has a handle decomposition with no 3-handles (which is not a weak assumption, but one that sometimes arises geometrically), then $\pi_1(N)$ surjects onto $\pi_1(M)$. So, if additionally $\pi_1(N)$ is finite (which is a very strong assumption), then $\pi_1(M)$ is finite. $\endgroup$ Jul 11, 2022 at 9:57

2 Answers 2

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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.

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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.

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    $\begingroup$ Your connect sum argument (with various other closed four-manifolds) also shows that the answer to the first part of question (1) is "no". That is, we cannot compute the fundamental group of a four-manifold just knowing its boundary. $\endgroup$
    – Sam Nead
    May 27, 2022 at 19:57

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