# Computation of $\pi_1$ for a Mazur manifold and its boundary

If we attach a $$4$$-dimensional $$1$$-handle $$D^1 \times D^3$$ to a $$4$$-dimensional $$0$$-handle $$B^4$$, we obtain $$S^1 \times D^3$$. The null homologous knot in $$S^1 \times S^2$$ indicated in the picture lives in a solid torus/attaching region $$S^1 \times D^2$$ and $$S^1 \times D^2 \hookrightarrow S^1 \times S^2 \hookrightarrow S^1 \times D^3.$$

In pg. 356 of his book Knots and Links, Dale Rolfsen notes that

1. $$\pi_1(W) = 1$$,
2. $$\pi_1(\partial W) = \langle x,y \ \vert \ y^5 = x^7, y^4 = x^2 y x^2 \rangle \neq 1$$.

How we compute the relevant fundamental groups using the following diagram? What is the strategy?

• Handle attachments are basically "structured" cw-complexes. There is a standard process to compute the fundamental group of CW-complexes described in algebraic topology textbooks. For the boundary there are a few ways. One would be to start with the Wirthinger presentation of the link exterior, then attach the appropriate relators as one fills. I think Rolfsen explains this, a little earlier in his book. Feb 19 at 0:19
• I don't understand. We can apply the Wirtinger's method to the knots in $S^3$. In OP, knots live in $S^1 \times S^2$. Is there a way generalize this process to another $3$-manifolds? Feb 19 at 10:46
• Yes, $S^1 \times S^2$ you can obtain as $0$-surgury on the unknot, so your surgury diagram for $\partial W$ will consist of a union of the surgery curves in your diagram with a $0$-labelled unknot. Going around the $S^1$ factor of $S^1 \times S^2$ corresponds to linking the unknot. Feb 19 at 19:57
• Great! To be hundred percent sure, would you sketch a diagram, please? Feb 19 at 23:01
• Plus, do we generalize your technique to any surgered $3$-manifold, or $S^1 \times S^2$ is special here? Do you have a reference (providing proofs) for generalizing process of Wirtinger presentations? Feb 19 at 23:03

Since $$1$$-handles and $$2$$-handles respectively give the generators and relations of the fundamental group of a $$4$$-manifold, we have (for my $$W$$): $$\pi_1(W) = \langle \alpha \ \vert \ \alpha^2 \alpha^{-1} \rangle = 1.$$
The boundary $$3$$-manifold $$\partial W$$ is already in the picture. Further, we can compute $$\pi_1(\partial W)$$ from the diagram by using Wirtinger's presentation.
Bonus: In particular, $$W$$ is a contractible $$4$$-manifold: By using Mayer-Vietoris sequences, observe that $$W$$ is a homology $$4$$-ball, i.e., we have $$H_*(W, \mathbb{Z}) = H_*(B^4, \mathbb{Z})$$. Then apply the classical theorems of Hurewicz and Whitehead.
Bonus 2: Since $$W$$ is contractible, $$\partial W$$ must be a homology $$3$$-sphere, i.e., $$H_*(\partial W, \mathbb{Z}) = H_*(S^3, \mathbb{Z})$$. Using Kirby calculus, show that $$\partial W \approx \Sigma(2,5,7)$$ where $$\Sigma(p,q,r)$$ denotes the Brieskorn sphere given coprime positive integers $$p,q$$ and $$r$$.