On the degree of global smoothings for subanalytic sets

In Bierstone and Parusiński (Duke Math J 167(16):3115–3128, 2018) proved the existence of global smoothings for closed subanalytic sets, both in an embedded and a non-embedded sense. In particular, in the non-embedded desingularization procedure the authors constructed smoothings of (generically) even degree, indeed it is well-known the existence of subanalytic sets which do not admit non-embedded smoothings of (generically) odd degree. In this paper we introduce a natural topological notion of nonbounding equator for subanalytic sets and prove a criterion to determine whether a closed subanalytic set X only admits global smoothings of even degree along the nonbounding equator. More in detail, we prove that if X has a nonbounding equator Y then every smoothing of X which is a covering on a connected neighborhood W of Y has even degree over W.