Self-maps of the product of two spheres fixing the diagonal

Abstract

We compute the monoid of essential self-maps of SⁿxSⁿ fixing the diagonal. More generally, we consider products SxS, where S is a suspension. Essential self-maps of SxS demonstrate the interplay between the pinching action for a mapping cone and the fundamental action on homotopy classes under a space. We compute examples with non-trivial fundamental actions.