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Topological types of Pfaffian manifolds. (English) Zbl 1074.14051

Let \(M\) be a \(C^2\) manifold, and \(\Omega= \{\omega_1,\dots, \omega_q\}\) a collection of \(C^1\) differential 1-forms on \(M\). A Pfaffian manifold of \(\Omega\) is a maximal connected \(C^1\) manifold \(V\) immersed in \(M\) such that \(T_xV= \bigcap^q_{i= 1}\ker(\omega_i)\) for any point \(x\) in \(V\). A Pfaffian manifold \((V,\Omega, M)\) is said to be of Rolle type if it verifies the following property: for any \(C^1\) path \(\gamma: [0, 1]\to M\) such that \(\gamma(0)\) and \(\gamma(1)\) are contained in \(V\), and for any \(i= 1,\dots,q\), there exists \(t_i\in [0, 1]\) such that \(\omega_i(\gamma(t_i))\cdot\gamma'(t_i)= 0\).
In the case when \(\Omega= \{\omega_1,\dots, \omega_q\}\) is a collection of differential 1-forms with polynomial coefficients in \(\mathbb{R}^n\), the authors prove that the number of homeomorphism classes of all Pfaffian manifolds of Rolle type of \(\Omega\) is finite and bounded by a computable function of \(n\), \(q\), and the degrees of \(\omega_1,\dots, \omega_q\). (The finiteness is proved in any \(o\)-minimal structure.)
The article also contains an example of a semi-algebraic \(C^1\) differential form \(\omega\) on a semi-algebraic \(C^2\) manifold of dimension 3 such that the set of homeomorphism classes of the Pfaffian manifolds of \(\omega\) has the cardinality of continuum.

MSC:

14P10 Semialgebraic sets and related spaces
03C64 Model theory of ordered structures; o-minimality
58A17 Pfaffian systems
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