Global Bounds for the Betti Numbers of Regular Fibers of Differentiable Mappings

IT is well known that the Betti numbers of any fiber p1 (~) of a polynomial mapping p: R" ~ R TM are bounded by some constants, depending only on n, m and the degree ofp (see e.g. [7] ). Now let f be a k times differentiable mapping of a bounded domain, with all the derivatives of order k bounded by a constant Mk. We can think of Mk as a measure of the deviation o f f from a polynomial mapping of degree k 1; as far as the deviation in a CLnorm is concerned, j ___ k 1, the Taylor formula gives the precise expression for it. The important general phenomenon is that also in much more delicate questions, concerning the topology and the geometry of the mapping f, its "deviation" from the "polynomial behavior" can be bounded in terms of M k. In [11] this fact was established for the structure of critical points and values off, and in [12] for some geometric properties of its fibers. The aim of the present paper is to extend in the same spirit to k-smooth mappings the property of polynomial ones, given above: the boundness of the Betti numbers of the fibers. Clearly it is impossible to bound the Betti numbers of each fiber: any closed set can be the set of zeroes ofa C~-smooth function. So the proper way to generalize the above property of polynomials is the following: First, we prove for any f the existence of fibers with the Betti numbers bounded by constants, depending only on M k (and, of course, on k and on the dimensions and the size of the domain and image of f ) . Secondly, we estimate, in the same terms, the integrals over the image oftbe Betti numbers of the fibers off. In particular, we answer a question concerning the conditions of integrability of the Banach indicatrix of a differentiable mapping, which was open for a long time (see [1], [2], [9]). All the inequalities below have the following form: they consist of a term, corresponding to the case of polynomials, and of a "correction term", containing the factor M k. Thus, for M k = 0, i.e. for f a polynomial of degree k 1, we obtain, up to constants, the usual bounds. The results below, as well as the results of [11] and [12] can be considered as the description of "the worst" possible behavior of k-smooth mappings. However, mainly they intend to answer another question: what can be said about the topology of a smooth or polynomial (of high degree) mapping, if the only information on its derivatives of order ~ k (where k is fixed and "small") we want to use, concerns their uniform bounds. Thus, we can reformulate most of results below (and of [11], [12] ) for polynomials only, without mentioning differentiable functions at all. In this setting they show how to work with polynomials of high degree, as if they were polynomials of low degree. Another important remark concerns the existence results below: in many cases we prove the existence of at least one value ~ in the image off, for which the Bctti numbers of the fiber f t (~) are bounded by suitable constants. Although we do not touch in this paper the question of explicitly finding such values, we should mention that the corresponding results can be brought to a rather effective form: for instance, we can prove that in any regular net with a sufficiently small (explicitly given) step, there are points ~ with the required properties. The author would like to thank the Max-Planck-Institut for Mathematik, where this paper was written, for its support and kind hospitality.