On the topology of the set of singularities of a solution to the Hamilton–Jacobi equation|Sur la topologie des singularités d'une solution de l'équation de Hamilton–Jacobi

We address the topology of the set of singularities of a solution to a Hamilton–Jacobi equation. For this, we will apply the idea of the first two authors (Cannarsa and Cheng, Generalized characteristics and Lax–Oleinik operators: global result, preprint, arXiv:1605. 07581, 2016) to use the positive Lax–Oleinik semi-group to propagate singularities