## 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

### Keywords:

Pfaffian manifold; Rolle type
Full Text:

### References:

 [1] DOI: 10.5802/aif.1297 · Zbl 0759.32005 · doi:10.5802/aif.1297 [2] Nash manifolds, Lecture Notes in Mathematics 1269 (1980) [3] Volumes, feuilles de Rolle et feuilletages analytiques réeles et théorème de Wilkie, Ann. Toulouse 7 pp 93– (1998) [4] Etude des hypersurfaces Pfaffiennes de Rolle (1991) [5] Soviet Math. Dokl. 22 pp 762– (1980) [6] DOI: 10.1007/BF02564582 · Zbl 0085.17303 · doi:10.1007/BF02564582 [7] J. Amer. Math. Soc. 9 pp 1051– (1996) [8] Springer Lecture Notes in Computer Science 33 pp 515– [9] DOI: 10.1215/S0012-7094-96-08416-1 · Zbl 0889.03025 · doi:10.1215/S0012-7094-96-08416-1 [10] DOI: 10.5802/aif.1629 · Zbl 0901.57029 · doi:10.5802/aif.1629 [11] Tame topology and O-minimal structure, London Math. Soc. Lecture Note 248 (1998) [12] DOI: 10.1007/s000140050025 · Zbl 0888.57026 · doi:10.1007/s000140050025 [13] J. reine angew. Math. 508 pp 189– (1999) [14] DOI: 10.1145/235809.235813 · Zbl 0885.68070 · doi:10.1145/235809.235813 [15] Geometry of subanalytic and semialgebraic sets (1997) · Zbl 0889.32006 [16] DOI: 10.1215/S0012-7094-00-10322-5 · Zbl 0970.32009 · doi:10.1215/S0012-7094-00-10322-5 [17] DOI: 10.1007/BF01232274 · Zbl 0769.58050 · doi:10.1007/BF01232274
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.