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