When given degrees generate the cohomology ring of a manifold?

Question

Let $M$ be a connected closed manifold of dimension $n$.

Suppose we have a subset $I\subset I_n=\{1, \dots, n\}$ such that for any two continuous maps $f, g:M\to M$ if $f^*=g^*|_{\oplus_{i\in I}H^i(M, \mathbb{Z})}$ then $f^*=g^*|_{\oplus_{i\in I_n}H^i(M, \mathbb{Z})}$.

Does it follow that $\oplus_{i\in I}H^i(M, \mathbb{Z})$ generates $\oplus_{i\in I_n}H^i(M, \mathbb{Z})$ as a graded ring?

Answer 1

I think $\mathbb{C}P^3$ is a counterexample. Since $\text{H}(\mathbb{C}P^3;\mathbb Z) = \mathbb Z[\alpha]/\alpha^4$ with $\alpha$ of degree $2$, we can take $I = \{6\}$, then if $f_{\ast} \alpha^3 = g_{\ast} \alpha^3$ we must have $f_{\ast} \alpha = g_{\ast} \beta$ and thus $f_{\ast} = g_{\ast}$ in all degrees. But clearly $\text{H}^6$ does not generate the whole cohomology of $\mathbb CP^3$.