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

TL;DR

This paper introduces the concepts of relative LS category and higher topological complexity of maps, providing new invariants and bounds, and explores their relationships with classical invariants in topology.

Contribution

It defines and studies the properties of relative LS category and higher topological complexity of maps, including cohomological bounds and relations to existing invariants.

Findings

Cohomological lower bounds for higher topological complexity

Comparison between topological complexity and LS category via degree one maps

Relationships established between invariants of connected manifolds

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.