  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 $S \subset
{\R}^{m+n}$, where $\R$ is a real closed field, is defined by a Boolean formula
with $s$ polynomials of degrees less than $d$, and $\pi: {\R}^{m+n} \to {\R}^n$
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.
