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We compute the monoid of essential self-maps of of the product of twon-spheres fixing the diagonal. More generally, we consider products S x S,where S is a suspension. Essential self-maps of S x S demonstrate the interplaybetween the pinching action for a mapping cone and the fundamental action onhomotopy classes under a space. We compute examples with non-trivialfundamental actions.

Self-Maps of the Product of Two Spheres Fixing the Diagonal