A note on universal maps (1975)
A map d of the  n-dimension E uclidean unit ball  Bn  into itself is called universal if every map of Bn  into itself agrees with d at at least one point. Theorem. Let d be a map of Bn  into itself, let  A= d1 (Sn-1) where  Sn-1 is the boundary of, Bn  and let  f  be the restriction of d to A. Then d is universal if and only if the homomorphism qenerated by f between the corresponding Ce ch cohomology  groups f* :  Hn-1 (Sn-1) → Hn-1 (A) is nontrivial.