On the number of homotopy types of fibres of a definable map

In this paper we prove a single exponential upper bound on the number of possible homotopy types of the fibres of a Pfaffian map in terms of the format of its graph. In particular, we show that if a semi-algebraic set S subset of Rm+n, where R is a real closed field, is defined by a Boolean formula with s polynomials of degree less than d, and pi : Rm+n -> R-n is the projection on a subspace, then the number of different homotopy types of fibres of pi does not exceed s(2(m+1)n)(2(m)nd)(O(nm)). As applications of our main results we prove single exponential bounds on the number of homotopy types of semi-algebraic sets defined by fewnomials, and by polynomials with bounded additive complexity. We also prove single exponential upper bounds on the radii of balls guaranteeing local contractibility for semi-algebraic sets defined by polynomials with integer coefficients.