A note on universal maps

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 (S^n-1) where  S^n-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 C_e ch cohomology  groups f* :  H^n-1 (S^n-1) → H^n-1 (A) is nontrivial.