# On the Number of Homotopy Types of Fibres of a Definable Map

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

EE/CS 3-180

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

π does not exceed (2

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).

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

π does not exceed (2

^{m}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).

MSC Code:

55Q70

Keywords: