# Homotopy

Monday, April 16, 2007 - 9:15am - 10:05am

Saugata Basu (Georgia Institute of Technology)

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 ⊂ ℝ

is defined by a Boolean formula with s polynomials of degrees less than d, and

π: (R)

(R)

on a subspace, then the number of different homotopy types of fibres of

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 ⊂ ℝ

^{m+n}is defined by a Boolean formula with s polynomials of degrees less than d, and

π: (R)

^{m+n}→(R)

^{n}is the projectionon a subspace, then the number of different homotopy types of fibres of