Topology of Singularities of Optimal Semicouplings

TL;DR

This paper investigates the topological structure of singularities in optimal semicouplings between spaces of unequal dimension, providing homotopy-reduction methods under certain geometric conditions.

Contribution

It introduces a framework for homotopy-reductions of source spaces onto singularity sets in optimal semicouplings with Riemannian targets, under the Uniform Halfspace condition.

Findings

Homotopy-reduction from source space to singularities

Construction of continuous strong deformation retracts

Applicability under Uniform Halfspace condition

Abstract

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).