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 SβRm+n, where R is a real closed field, is defined by a Boolean formula
with s polynomials of degrees less than d, and Ο:Rm+nβRn
is the projection on a subspace, then the number of different homotopy types of
fibres of Ο 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.Comment: Improved combinatorial complexit