Rational relative sectional category

TL;DR

This paper introduces an algebraic model for the rational relative sectional category of maps, linking it to cohomology for formal maps and exploring its algebraic properties in rational homotopy theory.

Contribution

It establishes a cohomology-based computation method for the rational relative sectional category of formal maps and extends algebraic characterizations to related invariants.

Findings

For formal maps, rational relative sectional category equals ideal nilpotency in cohomology.

The equality between these invariants may not hold in non-formal topological settings.

Provides algebraic descriptions for rational Lusternik-Schnirelmann category and higher topological complexity.

Abstract

We develop an algebraic model for the relative sectional category of a continuous map in rational homotopy theory using commutative differential graded algebras (CDGAs). Our main result establishes that for formal maps, the rational relative sectional category can be computed purely from cohomology, using ideal nilpotency. We also show that this equality may fail in general topological settings. Applying this framework, we obtain purely algebraic characterizations for the rational Lusternik-Schnirelmann category and the rational higher topological complexity of a map. Finally, we provide an algebraic description of the rational homotopic distance between formal maps.