Maps of degree one, relative LS category and higher topological complexities

Abstract: In this paper, we introduce relative LS category of a map and study some of its properties. Then we introduce higher topological complexity' of a map, a homotopy invariant. We give a cohomological lower bound and compare it with previously knowntopological complexity' of a map. Moreover, we study the relation between Lusternik-Schnirelmann category and topological complexity of two closed oriented manifolds connected by a degree one map.