Milnor's sphere $M$ is defined as the total space of the $S^3$ fiber bundle over $S^4$ with clutching map $f : S^3 \to SO(4)$ given by $u \mapsto (x \mapsto u^ixu^j)$, where $i, j$ are constants and $i + j = 1$. I am trying to compute $H_3(M)$ and $H_4(M)$. I can first decompose $M$ as $M = (D^+ \times S^3) \cap (D^- \times S^3)$, where $D^+, D^-$ are the upper and lower hemispheres of $S^4$. The Mayer-Vietoris sequence is given by
$$H_4(M) \to H_3(D^+\cap D^- \times S^3) \to H_3(D^+ \times S^3) \oplus H_3(D^- \times S^3) \to H_3(M)$$
I know $H_3(D^+\cap D^- \times S^3) = \mathbb Z \oplus \mathbb Z = H_3(D^+ \times S^3) \oplus H_3(D^- \times S^3)$, and I've shown that if the map between these is an isomorphism, then $H_4(M) = H_3(M) = 0$.
I'm stuck on showing this is an isomorphism. I understand the map is given by the induced homomorphisms on the inclusion maps, but I'm not sure how to explicitly construct it. My thought is to first find a basis on $H_3(D^+\cap D^- \times S^3) = \mathbb Z \oplus \mathbb Z$, but I'm not sure.