On the degree of global smoothings for subanalytic sets

In [4] Bierstone and Parusinski 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 construct 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 we 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 $W$.