[Paper Review] The monochromatic Hopf invariant

Guozhen Wang|arXiv (Cornell University)|Oct 27, 2014

Homotopy and Cohomology in Algebraic Topology16 references3 citations

TL;DR

This paper computes the monochromatic stable Hopf invariant of the β family at chromatic height two by analyzing the effect of the James-Hopf map on E₂-cohomology after applying the Bousfield-Kuhn functor to Morava E-theory. It shows that when a β element desuspends as expected, its stable Hopf invariant is β_{i-j/k}; otherwise, it has chromatic height at least three, resolving a conjecture from Bendersky et al.

ABSTRACT

In this paper we will compute the effect of the James-Hopf map after applying the Bousfield-Kuhn functor on Morava E-theory, and then compute the monochromatic Hopf invariant of the $β$ family using this cohomological information.

Motivation & Objective

• To understand the unstable homotopy of odd spheres at chromatic height two.
• To compute the monochromatic stable Hopf invariant of the β family using cohomological data from the Bousfield-Kuhn functor.
• To resolve the conjecture that the Hopf invariant of β_{i/j,k} should be β_{i-j/k} under desuspension.
• To clarify when the expected Hopf invariant holds and when higher chromatic height arises.

Proposed method

• Apply the Bousfield-Kuhn functor to the James-Hopf map and compute its effect on E₂-cohomology of Morava E-theory.
• Use the Adams-Novikov spectral sequence (ANSS) to analyze the E₂-term of the monochromatic layer M₂V(1) and M₂P₁.
• Leverage the K(1)-local isomorphism between the sphere spectrum and Σ^∞BΣ_p to relate the EHP sequence to the p-Bockstein spectral sequence.
• Analyze the filtration structure in the ANSS and AHSS to rule out higher differentials for p ≥ 5.
• Use the chromatic spectral sequence and unstable chromatic filtration to relate monochromatic invariants to actual stable Hopf invariants.
• Construct explicit desuspension maps using v₂-self-maps on V(1) to verify the Hopf invariant in favorable cases.

Experimental results

Research questions

• RQ1 Does the monochromatic Hopf invariant of β_{i/j,k} equal β_{i-j/k} when the element desuspends?
• RQ2 What is the chromatic height of the stable Hopf invariant when desuspension fails?
• RQ3 How do the E₂-cohomology computations of the Bousfield-Kuhn functor inform the monochromatic Hopf invariant?
• RQ4 Can the monochromatic invariant be lifted to the actual stable Hopf invariant using chromatic spectral sequences?
• RQ5 What role do filtrations in the ANSS and AHSS play in determining the leading term of the Hopf invariant?

Key findings

• For p ≥ 5, the ANSS for L_{K(2)}P₁ collapses at E₂, so all AHSS differentials arise from algebraic d₁-differentials.
• The monochromatic Hopf invariant of β₁ is β₀, represented by 1/(p v₁), and this matches the known stable Hopf invariant α₁.
• When β₁ desuspends to S^{2p-1}, the actual stable Hopf invariant is α₁, confirming the monochromatic computation.
• For k ≥ 2, β_k desuspends to S^{2p+3}, and its stable Hopf invariant is β_{k-1}, consistent with known results.
• In cases where β₁ cannot be extended to V(1), such as on S^{2p-1}, S^{2p+1}, or S^{2p+2}, the Hopf invariant is not β₀, indicating higher chromatic height.
• The method provides a lower bound on the sphere of origin for the Hopf invariant, with the actual invariant either matching the monochromatic computation or having chromatic height ≥ 3.