Rational approximations of sectional category and Poincar\'e duality

TL;DR

This paper extends the known equivalence between rational invariants and Poincaré duality complexes to the sectional category of fibrations, providing new insights into rational homotopy theory.

Contribution

It establishes an analogue of the equivalence between rational module category and Toomer invariant for the sectional category of fibrations.

Findings

Rational module category and Toomer invariant coincide for Poincaré duality complexes.

An analogue of this result is proven for the sectional category of fibrations.

Provides new tools for studying fibrations in rational homotopy theory.

Abstract

F\'elix, Halperin, and Lemaire have shown that the rational module category Mcat and the rational Toomer invariant $e_0$ coincide for simply connected Poincar\'e duality complexes. We establish an analogue of this result for the sectional category of a fibration.