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.
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