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On the stable Hopf invariant

John R. Klein

Abstract

We provide a simplified approach to the the stable Hopf invariant. We provide short elementary proofs of the Cartan Formula, the Composition Formula, and the Transfer formula. In addition, when $π$ is a discrete group, we show how to extend these results to the stable category of $π$-spaces.

On the stable Hopf invariant

Abstract

We provide a simplified approach to the the stable Hopf invariant. We provide short elementary proofs of the Cartan Formula, the Composition Formula, and the Transfer formula. In addition, when is a discrete group, we show how to extend these results to the stable category of -spaces.
Paper Structure (11 sections, 4 theorems, 78 equations)

This paper contains 11 sections, 4 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Relation to the Segal-Snaith operation
  3. Extension to $\pi$-spaces
  4. $\mathbb Z_2$-equivariant stable homotopy
  5. The tom Dieck splitting for $\mathbb Z_2$
  6. Specialization
  7. Definition of the stable Hopf invariant
  8. Proof of Theorem \ref{['bigthm:Hopf-properties']}
  9. The cup product again
  10. Proof of Theorem \ref{['bigthm:coincide']}
  11. Proof of Addendum \ref{['bigadd:Hopf-properties']}

Key Result

Theorem 1

There is an operation called the stable Hopf invariant that is natural in both $A$ and $B$ and which satisfies:

Theorems & Definitions (14)

  • Theorem 1
  • Remark 1.1
  • Theorem 2
  • Corollary 3
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3: $\mathbb Z_2$-equivariant Stable Maps
  • Remark 3.1
  • Definition 4.1
  • Lemma 5.1
  • ...and 4 more