On a lower bound in the problem of realization of cycles

One of the classical problems in topology is the Steenrod problem on realization of cycles: if $x\in H_n(X;\mathbb{Z})$ is an integral $n$-dimensional homology class of a topological space $X$, do there exist an oriented smooth $n$-dimensional manifold $M$ and a map $f\colon M \to X$ such that $x$ is the image of the generator of $H_n(M;\mathbb{Z})$. In 1954, R. Thom proved that an arbitrary $n$-dimensional integral homology class can be realized in the above sense after multiplication by some positive integer, which can be chosen uniformly for all $n$-dimensional classes. Let us denote by $k(n)$ the minimal of such positive integers. In my talk, I will give an overview of classical results about the Steenrod problem and tell my own results about boundaries for the number $k(n)$.