 Solution

Suppose such maps existed, and let $u$ generate $H^4(S^4;\mathbb{Z})$. Since $f^{\ast }{g}^{\ast }=1$, write $g^{\ast }u=ma$ and $f^{\ast }a=nu$; then $mn=1$, so $m=\pm 1$. Naturality of the cup product givescontradicting the nonzero degree-eight class computed in part c.