Topological invariants for singularities of real vector fields in dimension three
This paper is devoted to the determination of topological invariants of absolutely isolated singularities of three-dimensional real vector fields. Following the notation of the paper, if \(\xi_0\) is the germ of an analytic vector field over \((\mathbb{R}^3, 0)\) with an isolated singularity at the origin, the oriented foliation \(\mathcal{L}_0\) defined by \(\xi_0\) is given by any germ of vector field of the form \(f\xi_0\), where \(f\) is a germ of a positive analytic function. This foliation is said to have an \textit{absolutely isolated singularity} at the origin if and only if the transform of \(\mathcal{L}_0\) under any finite sequence of blowing-ups has only isolated singularities and the exceptional divisor is invariant. As it has been shown in [\textit{C. Camacho, F. Cano} and \textit{P. Sad}, Invent. Math. 98, No. 2, 351--369 (1989; Zbl 0689.32008)], the oriented foliations with an absolutely isolated singularity at the origin, have a canonical reduction of singularities \(\pi: (M,D) \to (\mathbb{R}^3,0)\) formed by the composition of a finite sequence of blowing-ups and such that the singularities of the corresponding vector field at \(M\) have a non-nilpotent linear part. Note that the exceptional divisor is given by \(D\), that \(\partial M=D\) and that \(D = \bigcup_{i=1}^{n} D_i\), where \(D_i\) is an irreducible component created at each blowing-up. Two germs \(\xi_0\) and \(\xi_0'\) of analytic vector fields over \((\mathbb{R}^3,0)\) are said to be \textit{\(\pi\)-topologically equivalent} if there is a homeomorphism \(h: (M,D) \to (M,D)\) that respects the natural stratification of \(M\) induced by the decomposition \(D = \bigcup_{i=1}^{n} D_i\) and that induces an oriented topological equivalence between the transforms \(\mathcal{L}\) and \(\mathcal{L}'\) of each oriented foliation defined by \(\xi_0\) and \(\xi_0'\), respectively. In this paper, a complete classification for the \(\pi\)-topological equivalence in a wide class \(\mathcal{H}_\pi\) of vector fields with absolutely isolated singularities and which admit \(\pi\) as a desingularization morphism, is given. This wide class \(\mathcal{H}_\pi\) of vector fields is defined to satisfy several generic properties such as \textit{hyperbolicity} of all the singularities of \(\mathcal{L}\) at \(M\), a \textit{non-return} property which avoids the existence of invariant cycles or oriented cycles (see the reviewed paper for a proper definition) and that only \textit{rigid saddle connections} are allowed. In fact, the complete classification for the \(\pi\)-topological equivalence is given in the general space \(\mathcal{H}(M,D)\) of oriented foliations by lines defined on \((M,D)\) and which satisfy the previously described generic conditions. Any topological equivalence must respect the \textit{critical locus} of \(\mathcal{L}\), denoted by \(\mathcal{C}(\mathcal{L})\). Thus, the space \(\mathcal{H}_{\mathcal{C}}(M,D)\) of foliations in \(\mathcal{H}(M,D)\) having a fixed critical locus is taken into account. Given a foliation \(\mathcal{L}\) in the latter space, a finite set of paths \(\mathcal{P}(\mathcal{L})\) formed with edges in \(\mathcal{C}\) and depending only on the eigenvalues in the singularities at \(\mathcal{L}\) is defined. Two foliations in \(\mathcal{H}_{\mathcal{C}}(M,D)\) that are topologically equivalent need to have the same set of paths \(\mathcal{P}\). The main result of the paper is the reciprocal of this fact. Theorem. Let \(\mathcal{L}, \mathcal{L}' \in \mathcal{H}_\mathcal{C}(M,D)\) and assume that \(\mathcal{P}(\mathcal{L}) =\mathcal{P}(\mathcal{L}')\). Then, there is a topological equivalence \(h: (M,D) \to (M,D)\) between \(\mathcal{L}\) and \(\mathcal{L}'\). The proof of this theorem is constructive and all the objects taken into account are defined step by step. Thus, this paper provides a source of new material to go on the study of the topological properties of foliations on \(\mathbb{R}^3\).
scientific article; zbMATH DE number 5347947