Equivariant weight filtration for real algebraic varieties with action

TL;DR

This paper develops an equivariant weight filtration for the homology of real algebraic varieties with finite group actions, revealing new invariants and connecting to existing equivariant Betti numbers.

Contribution

It introduces an equivariant weight filtration on equivariant homology, constructs additive invariants from it, and relates them to known equivariant virtual Betti numbers.

Findings

Constructed an equivariant weight filtration using group homology and the weight complex.

Established additive invariants from elementary bricks of the filtration.

Connected these invariants to Fichou's equivariant virtual Betti numbers in specific cases.

Abstract

We show the existence of an equivariant weight filtration on the equivariant homology of real algebraic varieties equipped with a finite group action, by applying group homology onto the weight complex of McCrory and Parusi\'nski. The group action enriches the information contained in the induced equivariant weight spectral sequence from which we can not extract finite additivity directly if the group is of even order. Still, we manage to construct additive invariants from its elementary bricks, and prove that they are induced from the very equivariant geometry of real algebraic varieties with action, and that they coincide with Fichou's equivariant virtual Betti numbers in some cases. In the case of the two-elements group, we recover these through the use of globally invariant chains and the equivariant version of Guill{\'e}n and Navarro Aznar's extension criterion.