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 ⊂ Rm+n, where R is a real closed field, is defined by a Boolean formula with s polynomials of degree less than d, and π : Rm+n → Rn is the projection on a subspace, then the number of different homotopy types of fibres of π does not exceed s2(m+1)n(2mnd)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.