# Killing cohomology classes from $\operatorname{BSO}(2n)$

We know that the mod-2 cohomology of $$\operatorname{BSO}(2n)$$ is the polynomial algebra of Stiefel-Whitney 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 $$\operatorname{Deg}(y)= 2n$$?

• There should be obstructions coming from Steenrod operations. – Qiaochu Yuan Sep 30 at 23:47
• 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) – Dylan Wilson Sep 30 at 23:57
• (Actually I guess showing n=4 fails requires a little more than just hopf invariant 1: ams.org/journals/proc/1963-014-01/S0002-9939-1963-0150763-5/…) – Dylan Wilson Oct 1 at 0:07