Bounding Betti Numbers of Sets Definable in O-Minimal Structures Over the Reals

TL;DR

This paper establishes bounds on Betti numbers of sets definable in o-minimal structures, generalizing classical results and providing new bounds for complex definable sets, including those with quantifiers and Pfaffian functions.

Contribution

It introduces an axiomatic complexity measure to unify bounds for various definable sets in o-minimal structures, extending classical bounds to more complex cases.

Findings

Provides a generalized Thom-Milnor bound for o-minimal sets.

Derives bounds for sets defined by Boolean combinations of equations and inequalities.

Establishes a singly exponential bound for sub-Pfaffian sets with fixed quantifier alternations.

Abstract

A bound for Betti numbers of sets definable in o-minimal structures is presented. An axiomatic complexity measure is defined, allowing various concrete complexity measures for definable functions to be covered. This includes common concrete measures such as the degree of polynomials, and complexity of Pfaffian functions. A generalisation of the Thom-Milnor Bound for sets defined by the conjunction of equations and non-strict inequalities is presented, in the new context of sets definable in o-minimal structures using the axiomatic complexity measure. Next bounds are produced for sets defined by Boolean combinations of equations and inequalities, through firstly considering sets defined by sign conditions, then using this to produce results for closed sets, and then making use of a construction to approximate any set defined by a Boolean combination of equations and inequalities by a closed set. Lastly, existing results for sets defined using quantifiers on an open or closed set are generalised, using a construction from Gabrielov and Vorobjov to approximate any set by a compact set. This results in a method to find a general bound for any set definable in an o-minimal structure in terms of the axiomatic complexity measure. As a consequence for the first time an upper bound for sub-Pfaffian sets defined by arbitrary formulae with quantifiers is given. This bound is singly exponential if the number of quantifier alternations is fixed.