Secondary cohomology operations and sectional category

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.

💡 Research Summary

The paper investigates lower bounds for the sectional category (secat) of a fibration q : E → B by exploiting secondary cohomology operations defined on the total space of the fibrewise join. Classical lower bounds for secat include the nilpotency of the kernel of the induced map in cohomology (nil ker q*) and the “module weight” wgt(q;R), which measures injectivity of the map induced by the k‑fold fibrewise join. Iwase and Kono refined this to the “module weight” Mwgt(q;𝔽ₚ), requiring a retraction that is a morphism of Aₚ‑modules (Steenrod algebra modules). However, Mwgt can still be strictly smaller than the true secat.

The authors introduce a new invariant, the secondary module weight Swgt(q;𝔽ₚ), based on secondary cohomology operations. Whenever two stable operations θ and φ satisfy a relation φ∘θ ≃ ∗ (for example the Adem relation Sq³Sq¹ + Sq²Sq² ≡ 0), a secondary operation Φ: ker θ → coker φ is defined. Φ is natural and commutes with all Aₚ‑module maps. Swgt(q;𝔽ₚ) is defined as the smallest k such that the map q(k)* : H⁎(B;𝔽ₚ) → H⁎(E(k);𝔽ₚ) admits a retraction that commutes with all primary and secondary operations. By construction Swgt ≥ Mwgt ≥ wgt ≥ nil ker, and the paper shows that the inequality can be strict.

Two concrete examples illustrate the strength of Swgt.

1. Twistor bundle q : ℂP⁵ → ℍP². The ordinary weight wgt(q;𝔽₂) is 0 because q* is injective, while Mwgt(q;𝔽₂) equals 1 (a retraction exists after one fibrewise join). The authors consider the 2‑fold fibrewise join q(1) and its total space E(1). Using the secondary operation Φ derived from the Adem relation, they compute Φ on the generator y ∈ H⁵(E(1);𝔽₂). By a careful Thom‑isomorphism argument and Harp­er’s “compatibility with exact sequences” technique, they show Φ(y) = q(1)*(a²) ≠ 0, where a² generates H⁸(ℍP²;𝔽₂). Since Φ vanishes on H⁵(ℍP²), any retraction commuting with Φ would force a² = 0, a contradiction. Hence Swgt(q;𝔽₂) = 2, matching the actual sectional category secat(q) = 2, while Mwgt and wgt are strictly smaller.

2. A 3‑sphere co‑fiber X obtained as the homotopy co‑fiber of the composition ω∘η : S⁶ → S³ (η ∈ π₇(S⁶), ω ∈ π₆(S³)). For this space cat(X) = 2. The primary weights wgt(X;𝔽₂) and Mwgt(X;𝔽₂) both equal 1. Again the same secondary operation Φ detects a non‑trivial element in H⁸ of the 2‑fold fibrewise join of the based path fibration over X, forcing Swgt(X;𝔽₂) = 2. The non‑triviality of Φ reflects the Hopf invariant of the attaching map of the top cell, which is represented by η∘η ∈ π₇(S⁵).

The proofs rely heavily on Harp­er’s method of computing secondary operations via cofiber sequences. For the twistor bundle, the authors identify the cofiber of the inclusion of the fibre into the fibrewise join with a Thom space of a sphere bundle, use the Thom isomorphism to relate Steenrod squares of the Thom class to Stiefel‑Whitney classes, and then apply the exact‑sequence compatibility to evaluate Φ. For the second example, they use known calculations of Steenrod squares in loop spaces of Spin(n) (Kono‑Kozima) to obtain the required non‑vanishing.

In summary, the paper defines a refined invariant Swgt that incorporates secondary cohomology operations, proves that it provides a strictly stronger lower bound for sectional category than previously known invariants, and demonstrates its effectiveness through explicit calculations. This work opens a new avenue for applying higher‑order cohomology operations to problems in Lusternik‑Schnirelmann theory, topological complexity, and related homotopical invariants.