I have problem understanding the proof of this theorem on page 23 of Kirby's Topology of 4-manifolds:
Two closed oriented 4-manifolds are homotopy equivalent iff they have isomorphic intersection forms.
They consider a map $S^3\to S^2$v...v$S^2$ and the inverse image of $p_i$ (a point in $i$-th $S^2$) which are knots in $S^3$ and claim this map is determined up to homotopy by the linking matrix of these knots in $S^3$. But I cannot figure out why such map is determined by this linking matrix up to homotopy. Any help?