Remarks on filtrations of the homology of real varieties

TL;DR

This paper disproves Teh's conjecture relating niveau filtration and cycle maps in real varieties, showing the filtration's triviality and providing explicit counterexamples, thereby clarifying the relationship between various cycle maps.

Contribution

The paper demonstrates the falsehood of Teh's conjecture, establishes the triviality of the niveau filtration on reduced Lawson homology, and clarifies the factorization of cycle maps.

Findings

Niveau filtration on reduced Lawson homology is trivial.

Teh's conjecture relating niveau filtration and cycle maps is false.

Borel-Haeflinger cycle map factors through reduced Lawson homology cycle map.

Abstract

We demonstrate that a conjecture of Teh which relates the niveau filtration on Borel-Moore homology of real varieties and the images of generalized cycle maps from reduced Lawson homology is false. We show that the niveau filtration on reduced Lawson homology is trivial and construct an explicit class of examples for which Teh's conjecture fails by generalizing a result of Schulting. We compare various cycle maps and in particular we show that the Borel-Haeflinger cycle map naturally factors through the reduced Lawson homology cycle map.