We know that the mod2 cohomology of $\operatorname{BSO}(2n)$ is the polynomial algebra of StiefelWhitney classes $w_i$ with $\mathbb Z/2$coefficients, $2\le i\le 2n$. Is it possible to kill off arbitrary cohomology classes to obtain a certain space with the desired cohomology? For example can one kill cohomology classes from $\operatorname{BSO}(2n)$ to form a space $Y$ such that $H^*(Y)$ is the polynomial algebra of one generator $y$ with $\operatorname{Deg}(y)= 2n$?
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3$\begingroup$ There should be obstructions coming from Steenrod operations. $\endgroup$ – Qiaochu Yuan Sep 30 at 23:47

6$\begingroup$ this is not possible as soon as n is larger than 2 in fact no space Y exists with mod 2 cohomology polynomial on a class of degree not equal to 1, 2, or 4 (This follows from the solution to the hopf invariant 1 problem) $\endgroup$ – Dylan Wilson Sep 30 at 23:57

3$\begingroup$ (Actually I guess showing n=4 fails requires a little more than just hopf invariant 1: ams.org/journals/proc/196301401/S00029939196301507635/…) $\endgroup$ – Dylan Wilson Oct 1 at 0:07