Splitting homotopy idempotents which have essential fixed points

The homomorphism on fundamental group induced by a homotopy idempotent having essential fixed points

“…The background material for that problem is disjoint from the background material for Theorem 1.2. Hence we have separated the two: that application is made in our paper [3], which is intended to be a sequel to the present paper. However, to give a context for Theorem 1.2, we will say where homotopy idempotents actually arise.…”