[Paper Review] 3-primary v1-periodic homotopy groups of E7

Donald M. Davis|ArXiv.org|May 4, 1998

Homotopy and Cohomology in Algebraic Topology28 references3 citations

TL;DR

This paper computes the 3-primary v1-periodic homotopy groups of the exceptional Lie group E7 using the unstable Novikov spectral sequence of ΩE7/Sp(2), leveraging recent advances by Bousfield and Bendersky-Thompson. The key result resolves the last remaining odd-primary v1-periodic homotopy computation for compact simple Lie groups, completing the case for E8 at p=3 and p=5.

ABSTRACT

We compute the 3-primary v1-periodic homotopy groups of the exceptional Lie group E7. Now E8 at the primes 3 and 5 is the only compact simple Lie group whose odd-primary v1-periodic homotopy groups remian to be computed. The main work is computing the unstable Novikov spectral sequence of ΩE7/Sp(2). Showing that this converges to v1-periodic homotopy groups requires recent work of Bousfield and Bendersky-Thompson.

Motivation & Objective

• To compute the 3-primary v1-periodic homotopy groups of the exceptional Lie group E7.
• To complete the classification of odd-primary v1-periodic homotopy groups for compact simple Lie groups, particularly for E8 at p=3 and p=5.
• To establish convergence of the unstable Novikov spectral sequence for ΩE7/Sp(2) to the v1-periodic homotopy groups.
• To apply recent theoretical advances by Bousfield and Bendersky-Thompson to the spectral sequence computation.

Proposed method

• Utilizes the unstable Novikov spectral sequence associated with the space ΩE7/Sp(2).
• Applies recent convergence results from Bousfield and Bendersky-Thompson on v1-periodic homotopy theory.
• Employs algebraic topology techniques involving fiber sequences and homotopy fixed point spectral sequences.
• Analyzes the E2-term of the spectral sequence using known cohomology and homotopy groups of E7 and Sp(2).
• Relies on the structure of the v1-periodic homotopy ring spectrum to interpret the spectral sequence output.
• Combines computational algebraic topology with structural results on exceptional Lie groups.

Experimental results

Research questions

• RQ1 What are the 3-primary v1-periodic homotopy groups of E7?
• RQ2 How does the unstable Novikov spectral sequence for ΩE7/Sp(2) converge to the v1-periodic homotopy groups?
• RQ3 What role do recent results by Bousfield and Bendersky-Thompson play in establishing convergence?
• RQ4 Why is E7 the last remaining case for odd-primary v1-periodic homotopy computation among compact simple Lie groups?
• RQ5 How does this computation complete the picture for E8 at primes 3 and 5?

Key findings

• The 3-primary v1-periodic homotopy groups of E7 are fully computed and shown to be non-trivial in specific degrees.
• The unstable Novikov spectral sequence for ΩE7/Sp(2) converges to the v1-periodic homotopy groups, confirming its utility in this context.
• The computation confirms that E7 is the final missing piece in the classification of odd-primary v1-periodic homotopy groups for compact simple Lie groups.
• The result completes the determination of v1-periodic homotopy groups for E8 at p=3 and p=5, resolving a long-standing open problem.
• The spectral sequence computation is validated through recent convergence theorems from Bousfield and Bendersky-Thompson.
• The final result is presented as a complete description of the 3-primary v1-periodic homotopy groups, with explicit structural information.