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Let $X$ be a space and $I$ be an ideal in the cohomology ring $H^*(X)$.

I am interested in the question whether there is a map of spaces $Y\rightarrow X$ such that $I$ is the kernel of the induced map $H^*(X)\to H^*(Y)$.

For example, that ideal needs to be closed under the cohomology operation of $H^*$, e.g., if we work with singular cohomology with $\mathbb{F}_2$-coefficients, it needs to be closed under the Steenrod operations.

But somehow it feels like these are not the only obstructions and there should be much more obstructions coming from higher cohomology operations, but I can't see how to make this precise.

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    $\begingroup$ I wouldn't expect for any definitive answer to this question. You've stated some necessary conditions, but then you have more, like secondary operations, etc. $\endgroup$
    – Fernando Muro
    May 19, 2022 at 13:11
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    $\begingroup$ Suppose $I$ is generated by classes $x_i$ of degree $d_i$, represented by maps into Eilenberg-MacLane spaces $x_i:X\to K(R,d_i)$. Let $f:Y\to X$ be the inclusion of the homotopy fibre of the map $\prod_i x_i: X\to \prod_i K(R,d_i)$. The certainly $\ker(f^*)$ contains $I$ (but probably it's much larger?). $\endgroup$
    – Mark Grant
    May 19, 2022 at 13:51

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