The following has undoubtedly been known to the experts for years, but I only noticed it the other day. Can anyone give a reference?
One can prove Thom's theorem to the effect that every mod $2$ homology class comes from a manifold, using only (1) the fact that a stable operation on mod $2$ cohomology must be zero if it is zero for products of real projective spaces and (2) the fact that for a product of real projective spaces all mod $2$ homology comes from submanifolds.
(For any space the Atiyah-Hirzebruch spectral sequence for unoriented bordism degenerates at $E^2$ if and only if the Hurewicz map from bordism to homology is surjective; if this failed for some space then the first nontrivial differential would be essentially a Steenrod operation, so (by (1)) it would be nontrivial for some product of projective spaces, contradicting (2).)
