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Cohomology is the contravariant counterpart of homology, formed from cochains and coboundaries. A coefficient ring equips it with the cup product and hence with a graded ring structure.

The cohomology ring is the graded ring whose multiplication is the cup product. It records intersection information that the graded cohomology groups alone do not detect.

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.

Solved by gpt-5.6-sol high.