Suppose we have $\vee_{i=0}^k S^2$ the two skeleton and $h:S^3\to \vee_{i=0}^k S^2$ the attaching map to attach cell $e^4$.
My question is, if two attaching map $h_1$ and $h_2$ are homotopy, can we show that the CW complex induced by $h_1$ and $h_2$ have the same homotopy type?
Since the homotopy group is isomorphic automatically, I think all I need to do is to construct a map from the CW complex induced by $h_1$ to the CW complex induced by $h_2$, but I fail to construct this map. Can anyone help me? Thanks!