Approximation of definable sets by compact families, and upper bounds on homotopy and homology

TL;DR

This paper establishes new upper bounds on the homotopy and homology groups of o-minimal sets, improving Betti number bounds for semialgebraic and sub-Pfaffian sets using approximation techniques.

Contribution

It introduces novel upper bounds on topological invariants of o-minimal sets, including the first singly exponential bounds for sub-Pfaffian sets.

Findings

Improved upper bounds on Betti numbers of semialgebraic sets.

First singly exponential bounds on Betti numbers of sub-Pfaffian sets.

Enhanced understanding of the topology of definable sets via approximation.

Abstract

We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on Betti numbers of semialgebraic sets defined by quantifier-free formulae, and obtain for the first time a singly exponential bound on Betti numbers of sub-Pfaffian sets.