Secondary cohomology operations and sectional category

TL;DR

This paper introduces secondary cohomology operations to improve lower bounds on the sectional category of fibrations, refining existing invariants and providing examples where the new measure detects properties missed by previous methods.

Contribution

It proposes the secondary module weight as a refined invariant for sectional category, enhancing the detection capabilities beyond existing module weight measures.

Findings

Secondary module weight can detect sectional category where module weight cannot.

Examples demonstrate the effectiveness of secondary module weight at prime 2.

The approach refines the understanding of fibrations through cohomology operations.

Abstract

We show how secondary cohomology operations in the total space of the fibred join can be used to give lower bounds for the sectional category of a fibration. This suggests a refinement of the module weight of Iwase--Kono, which we call the secondary module weight. Examples are given for which the secondary module weight at the prime $2$ detects sectional category while the module weight does not.