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2004 Topological types of Pfaffian manifolds
Masato Fujita, Masahiro Shiota
Nagoya Math. J. 173: 1-22 (2004).


Let $\Omega = (\omega_{1}, \dots, \omega_{n-k})$ be differential $1$-forms with polynomial coefficients in ${\bf R}^{n}$. A Pfaffian manifold of $\Omega$ is by definition a maximal integral $k$-manifold of $\Omega$. It is shown that the number of homeomorphism classes of all Pfaffian manifolds of Rolle Type of $\Omega$ is finite and, moreover, bounded by a computable function in variables $n$, $k$ and the degree of $\omega_{1}, \dots, \omega_{n-k}$. Finiteness is proved also in any o-minimal structure.

We give also an example of a semi-algebraic $C^{1}$ differential form on a semialgebraic $C^{2}$ $3$-manifold whose Pfaffian manifolds have homeomorphism classes of the cardinality of continuum. Hence the cardinality of all manifolds is the continuum (not countable).


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Masato Fujita. Masahiro Shiota. "Topological types of Pfaffian manifolds." Nagoya Math. J. 173 1 - 22, 2004.


Published: 2004
First available in Project Euclid: 27 April 2005

zbMATH: 1074.14051
MathSciNet: MR2041754

Primary: 14P10
Secondary: 03C64

Rights: Copyright © 2004 Editorial Board, Nagoya Mathematical Journal

Vol.173 • 2004
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