The weight filtration for real algebraic varieties II: Classical homology

TL;DR

This paper introduces a weight filtration for classical homology of real algebraic varieties, extending previous work on Borel-Moore homology to provide a new algebraic-topological tool.

Contribution

It defines a filtered chain complex for real algebraic varieties that induces a weight filtration on classical homology, complementing earlier Borel-Moore homology results.

Findings

Defines the weight complex up to filtered quasi-isomorphism

Induces a weight filtration on classical homology with Z/2 coefficients

Extends the algebraic-topological framework for real algebraic varieties

Abstract

We associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on classical (compactly supported) homology with Z/2 coefficients an analog of the weight filtration for complex algebraic varieties. This complements our previous definition of the weight filtration of Borel-Moore homology.