# 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 $$N$$,

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!

• 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. May 27, 2022 at 0:41
• 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. Jul 11, 2022 at 9:57

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.

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.

• 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. May 27, 2022 at 19:57