Let $X$ be a compact manifold of dimension $\geq k$. Denote by \begin{equation} h: \pi _k(X) \rightarrow H_k(X,\mathbb{Z}) \end{equation} be Hurewicz homomorphism and by $\Gamma _k(X)\subset H_k(X,\mathbb{Z})$ its image. I want to look at the pairing \begin{equation} \langle \ . \ ,\ . \ \rangle : \ \Gamma _k(X)_{\text{free}} \times H^k(X,\mathbb{Z})_{\text{free}} \rightarrow \mathbb{Z} \end{equation} obtained by restricting the the Kronecker pairing between the free subgroups of homology and cohomology, which is non-degenerate. In general the restriction above will be degenerate. Indeed given a non-vanishing class $\omega \in H^k (X,\mathbb{Z})_{\text{free}}$, it is possible that for any $f: S^k \rightarrow X$ the pull-back $f^* \omega$ is zero. I'm looking for some cases in which I can say something more on this degeneracy.

In particular if $\omega \in H^k(X,\mathbb{Z})_{\text{free}}$ is also a generator of the cohomology ring (so that it cannot be written as the product of two classes of lower degree) I have some intuition that maybe $\langle \Sigma , \omega \rangle \neq 0$ for at least one $\Sigma \in \Gamma _k (X)$. I am not sure whether this is true, and if it is how to prove it. Or if it is false how to modify a bit statement to get some true fact connecting the generators of the cohomology ring with the image of the Hurewicz map. Can somebody say something about this?

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