Coincidence points of fiber maps on [formula omitted]-bundles

Abstract

In this note we study coincidence of pairs of fiber-preserving maps f , g : E 1 → E 2 where E 1 , E 2 are S n -bundles over a space B. We will show that for each homotopy class [ f ] of fiber-preserving maps over B, there is only one homotopy class [ g ] such that the pair ( f , g ) , where [ g ] = [ τ ○ f ] can be deformed to a coincidence free pair. Here τ : E 2 → E 2 is a fiber-preserving map which is fixed point free. In the case where the base is S 1 we classify the bundles, the homotopy classes of maps over S 1 and the pairs which can be deformed to coincidence free. At the end we discuss the self-coincidence problem.