I will describe some results giving 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 is defined by a Boolean formula with s polynomials of degrees less than d, and π: (R)m+n → (R)n is the projection on a subspace, then the number of different homotopy types of fibres of π does not exceed (2m snd)O(nm). All previously known bounds were doubly exponential. As applications of our main results we prove single exponential bounds on the number of homotopy types of semi-algebraic sets defined by polynomials having a fixed number of monomials in their support (we need to fix only the number of monomials, not the support set itself), as well as 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. (Joint work with N. Vorobjov).