On the topology of the set of singularities of a solution to the Hamilton-Jacobi equation

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. (C) 2016 Academie des sciences. Published by Elsevier Masson SAS.