Virtual Betti numbers of real algebraic varieties

TL;DR

This paper introduces virtual Betti numbers for real algebraic varieties, extending classical Betti numbers to all varieties while maintaining key additive properties, based on the weak factorization theorem.

Contribution

It establishes a unique extension of Betti numbers to all real algebraic varieties using the weak factorization theorem, preserving their additive nature.

Findings

Virtual Betti numbers are uniquely defined for all real algebraic varieties.

The extension preserves the additive property over subvarieties.

The approach relies on the weak factorization theorem.

Abstract

The weak factorization theorem for birational maps is used to prove that for all nonnegative i the ith mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a "virtual Betti number" beta_i defined for all real algebraic varieties, such that if Y is a closed subvariety of X, then beta_i(X) = beta_i(X\Y) + beta_i(Y).