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

TL;DR

This paper characterizes the monoid of essential self-maps of the product of two spheres fixing the diagonal, revealing complex interactions between homotopy actions and mapping cone pinching.

Contribution

It computes the monoid of essential self-maps fixing the diagonal for products of spheres, including cases with non-trivial fundamental actions, extending understanding of self-maps in homotopy theory.

Findings

Computed monoid of essential self-maps fixing the diagonal

Identified examples with non-trivial fundamental actions

Explored interplay between pinching actions and fundamental actions

Abstract

We compute the monoid of essential self-maps of of the product of two n-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 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.