Realizability of the $H_k$-distance functions by homology classes of path spaces

In the previous papers we constructed and studied mappings $d_k\colon M\times M\to\mathbb R$; we called them the $H_k$-distance functions. The main result of this paper is a theorem about the realizability of generalized distances $d_k(v,w)$, $v,w\in M$, considered as critical values of the length functional $\mathcal L\colon\Omega(M,v,w)\to\mathbb R$ generated by some nontrivial homology classes of the space $\Omega(M,v,w)$ of paths between points $v$ and $w$.