Diagonal Involutions and the Borsuk–Ulam Property for Product of Surfaces

In this work we study a generalization of the Borsuk–Ulam Theorem. Namely, we replace the sphere S n by a product of two closed surfaces M 2 × N 2 equipped with the diagonal involution T × S where T and S are free involutions on M 2 and N 2 , respectively, and the indexes i ( M 2 , T ) = i ( N 2 , S ) = 2 . Then we compute the index of the pair ( M 2 × N 2 , T × S ) and we obtain a Borsuk-Ulam Theorem for M 2 × N 2 .