Coepi maps and generalizations of the Hopf extension theorem

The Hopf extension theorem states that a map on the unit sphere in Rn is essential (i.e., each continuous extension to the unit ball has a zero) if and only if it has nonzero rotation (degree). We formulate and prove a corresponding result for coincidence points of condensing pairs of maps in infinite-dimensional spaces. To this end, a theory of coepi maps is introduced which in some sense is dual to the known theory of 0-epi maps. Also a uniqueness result for the coincidence index is obtained which provides a way to effectively calculate the index.