gn.general topology - Cohomology and fundamental classes

This is a reply to Alon's comment, but it's too long to be a comment and is probably interesting enough to be an answer.

Here's an example Thom gives of a homology class that is not realized by a submanifold: let $X=S^7/mathbb Z_3$, with $mathbb Z_3$ acting by rotations, and $Y=X times X$.
Then $H^1(X;mathbb Z_3)=H^2(X;mathbb Z_3)=mathbb Z_3$ (and they are related by a Bockstein); let $u$ generate $H^1$ and $v=beta u$ be the corresponding generator of $H^2$. Then it can be shown that the class $u otimes vu^2 - v otimes u^3 in H^7(Y;Z_3)$ is actually integral (i.e., in $H^7(Y;Z)$), and its Poincare dual in $H_7$ cannot be realized by a submanifold (in fact, it can't be realized by any map from a closed manifold to $Y$, which need not be the inclusion of a submanifold). This is a natural example to consider because the first obstruction to classes being realized by submanifolds comes from a mod 3 Steenrod operation, and these are easy to compute on $Y$ because $X$ is the 7-skeleton of a $K(mathbb Z_3,1)$. Note that the class in question is 3-torsion, so trivially 3 times it is realized by a submanifold.