Global bounds for the Betti numbers of regular fibers of differentiable mappings. (English) Zbl 0566.57014

Let f be a k times differentiable mapping of a bounded domain, with all the derivatives of order k bounded by a constant \(M_ k\), which can be taken to be a measure of the deviation of f from a polynomial mapping of degree k-1. In two earlier papers the author established the fact that \(M_ k\) also reflects the deviation of f from polynomial behaviour with respect to more delicate questions involving the geometry and topology of f, namely the structure of critical points and values of f and some geometric properties of its fibers.
In the present paper the author extends in the same spirit the following well-known property of polynomial mappings to k-smooth ones: the Betti numbers of any fiber of a polynomial mapping \(p: {\mathbb{R}}^ n\to {\mathbb{R}}^ m\) are bounded by constants depending only on m, n, and the degreee of p. The author first proves the existence of fibers with Betti numbers bounded by constants depending only on \(M_ k\), and then estimates the integrals over the image of the Betti numbers.
Reviewer: R.von Randow


57R35 Differentiable mappings in differential topology
57R70 Critical points and critical submanifolds in differential topology
26B99 Functions of several variables
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