Homology group and homotopy group of the standard twin

Question

Given a 4-sphere, if we cut out a solid 3-torus $B^2 \times S^1 \times S^1$ from a 4-sphere $S^4$ (with an unknotted torus), the remained exterior is called "the standard twin," say $M$.

What are the homology group $$H_n(M)$$ and homotopy group $$\pi_n(M)$$ of the standard twin?

Answer 1

By Alexander Duality,

$$\widetilde{H }_q(M) = \widetilde{H}^{3-q}(B^2\times S^1\times S^1) = \widetilde{H}^{3-q}(S^1\times S^1) = \begin{cases} \mathbb{Z} & q = 1\\ \mathbb{Z}^2 & q = 2\\ 0 & \text{otherwise}. \end{cases}$$

Therefore

$$H_q(M) = \begin{cases} \mathbb{Z} & q = 0, 1\\ \mathbb{Z}^2 & q = 2\\ 0 & \text{otherwise}. \end{cases}$$

I don't know how to calculate the homotopy groups of $M$.