Solution
= Solution
Suppose such maps existed, and let $u$ generate $H^4(S^4;\mathbb Z)$. Since $f^*g^*=1$, write $g^*u=ma$ and $f^*a=nu$; then $mn=1$, so $m=\pm1$. Naturality of the <cup product> gives
$$
0=g^*(u^2)=(g^*u)^2=a^2=d,
$$
contradicting the nonzero degree-eight class computed in part c.