An element $\alpha \in H_k(X; \mathbb{Z})$ is said to be *realisable* if there is a $k$-dimensional connected, closed, orientable $k$-dimensional submanifold $Y$ such that $\alpha = i_*[Y]$. The Steenrod Problem was whether every class is realisable. Thom showed in 1954 that this is not true in general. However, it is true if $0 \leq k \leq 6$. What is true however is that for any class $\alpha$, there is an integer $N$ such that $N\alpha$ is realisable.

We could ask the analogue of Steenrod's Problem for $\mathbb{Z}_2$ coefficients: is every class in $H_k(X; \mathbb{Z}_2)$ realisable? The answer is yes.

What if we ask the analogue of Steenrod's Problem for $\mathbb{Z}_p$ coefficients, for integer $p>2$?

Edit: Following Mark's comment, I am asking about realizability by smooth maps $i:Y \rightarrow X$.

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