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