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 known 'topological 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.

On the targeted complexity of a map

“…Recall that the higher complexity of a map is a generalization of usual complexity, which increases endpoints to more than two endpoints. In other words, the nth higher complexity considers paths as n pieces attached together; see [2,6,13]. In [6, Definition 3.7], the n-dimensional higher topological complexity of f was introduced for fibration maps.…”

“…Pavešic [10,11], Rami and Derfoufi [12], Murillo and Wu [8] defined and studied the topological complexities of maps, each with its own advantages, applications, and lacks. The topological complexity of maps was generalized to the higher topological complexity of maps denoted by T C n (f ); for more information, see [6,13].…”

Surgery Applications to a Generalized Rudyak Conjecture

“…In a recent paper [13], a new version of Rudyak's conjecture was considered in the context of topological complexity, but the present work first considers an even more general version of the conjecture for sectional category. Given a fibration p : E → B, the sectional category of p, denoted secat (p), is the smallest integer k such that B can be covered by k + 1 open sets U 0 , .…”