Codimension 2 cycles on products of projective homogeneous surfaces

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In the present paper, we provide general bounds for the torsion in the codimension 2 Chow groups of the products of projective homogeneous surfaces. In particular, we determine the torsion for the product of four Pfister quadric surfaces and the maximal torsion for the product of three Severi-Brauer surfaces. We also find an upper bound for the torsion of the product of three quadric surfaces with the same discriminant.