  We study the topology of singularities of $c$-optimal semicouplings in
unequal dimension. Our main results describe homotopy-reductions from a source
space $(X,\sigma)$ onto the singularities $Z_j$, $j\geq 0$ of $c$-optimal
semicouplings whenever $(Y, \tau)$ is a Riemannian target space and $c$ is a
cost on $X\times Y$ satisfying some general assumptions (A0)--(A5). We
construct continuous strong deformation retracts $X\leadsto Z_j$ whenever a
condition called Uniform Halfspace (UHS) condition is satisfied along
appropriate subsets. This article summarizes some results from the author's PhD
thesis (2019).
