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

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