Complexe de poids des vari\'et\'es alg\'ebriques r\'eelles avec action

TL;DR

This paper extends the weight complex framework to real algebraic varieties with finite group actions, establishing a filtered Smith sequence that accounts for symmetries and involutions.

Contribution

It introduces a weight complex with action for real algebraic varieties and proves a filtered Smith sequence incorporating Nash-constructible filtration.

Findings

Defined a weight complex with group action on real algebraic varieties.

Established a filtered Smith short sequence considering Nash-constructible filtration.

Proved the exactness of the sequence via splitting of Nash manifolds with involution.

Abstract

Using the functoriality of C. McCrory and A. Parusi\'nski's weight complex -which induces an analog of the weight filtration for complex algebraic varieties on the Borel-Moore homology with $\mathbb{Z}_2$ coefficients of real algebraic varieties-, we define a weight complex with action on the real algebraic varieties equipped with a finite group action. Emphasizing on the two elements group, we then establish a filtered version of Smith short sequence, taking into account the Nash-constructible filtration which realizes the weight complex with action. Its exactness is implied by the splitting of a Nash manifold equipped with an algebraic involution along an arc-symmetric subset.