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Apr 22, 2014 at 16:48 comment added nsrt You are asking whether $Z/2$-cohomology as a functor from the homotopy category of spaces to $Z/2$-algebras is full on certain morphism sets. This is wrong in general. First there is more structure on the algebraic invariants (they are algebras over the Steenrod algebra). Even with this modification, this is wrong in general, without assumptions on the spaces, as additional invariants can show. $BO(n)$ is not so special here.
Apr 22, 2014 at 12:29 answer added Mark Grant timeline score: 14
Apr 22, 2014 at 8:42 vote accept Oliver Straser
Apr 22, 2014 at 8:38 answer added Oscar Randal-Williams timeline score: 15
Apr 22, 2014 at 7:59 history asked Oliver Straser CC BY-SA 3.0