Self-maps of the product of two spheres fixing the diagonal (Q719529)

The authors investigate self-maps \(f:S^n\times S^n\to S^n\times S^n\) fixing the diagonal \(\Delta:S^n\to S^n\times S^n\), that is, with \(f\Delta=\Delta\), and the set \([S^n\times S^n,S^n\times S^n]^{\Delta}\) of homotopy classes of such maps, admitting only homotopies that fixes the diagonal. More generally, they consider products \(S\times S\), where \(S\) is a suspension, and study the interplay between the pinching action for a mapping cone and the fundamental action on homotopy classes under a space. They compute the orbits of the fundamental action for \(S^n\times S^n\) apparently providing the first example in the literature where such orbits are computed for subspaces other than points.