Nagoya Mathematical Journal

Topological types of Pfaffian manifolds

Masato Fujita and Masahiro Shiota

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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).

Article information

Nagoya Math. J., Volume 173 (2004), 1-22.

First available in Project Euclid: 27 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14P10: Semialgebraic sets and related spaces
Secondary: 03C64: Model theory of ordered structures; o-minimality


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

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