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Linear actions of $\mathbb{Z}/p\times\mathbb{Z}/p$ on $S^{2n-1}\times S^{2n-1}$

Jim Fowler, Courtney Thatcher

Abstract

For an odd prime $p$, we consider free actions of $(\mathbb{Z}/p)^2$ on $S^{2n-1}\times S^{2n-1}$ given by linear actions of $(\mathbb{Z}/p)^2$ on $\mathbb{R}^{4n}$. Simple examples include a lens space cross a lens space, but $k$-invariant calculations show that other quotients exist. Using the tools of Postnikov towers and surgery theory, the quotients are classified up to homotopy by the $k$-invariants and up to homeomorphism by the Pontrjagin classes. We will present these results and demonstrate how to calculate the $k$-invariants and the Pontrjagin classes from the rotation numbers.

Linear actions of $\mathbb{Z}/p\times\mathbb{Z}/p$ on $S^{2n-1}\times S^{2n-1}$

Abstract

For an odd prime , we consider free actions of on given by linear actions of on . Simple examples include a lens space cross a lens space, but -invariant calculations show that other quotients exist. Using the tools of Postnikov towers and surgery theory, the quotients are classified up to homotopy by the -invariants and up to homeomorphism by the Pontrjagin classes. We will present these results and demonstrate how to calculate the -invariants and the Pontrjagin classes from the rotation numbers.
Paper Structure (12 sections, 14 theorems, 45 equations)

This paper contains 12 sections, 14 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Constructions
  3. Notation
  4. Homotopy classification
  5. The first $k$-invariant and the homotopy type
  6. Simple homotopy classification
  7. Homeomorphism classification
  8. Classification by characteristic classes
  9. Surgery
  10. Normal invariants
  11. The Structure Set
  12. Applications and Open Questions

Key Result

Proposition 2.1

A representation of $(\mathbb Z_{/p})^2$ on $\mathbb R^{2n} \times \mathbb R^{2n}$ preserving the decomposition of $\mathbb R^{4n}$ into $\mathbb R^{2n} \times \mathbb R^{2n}$ is equivalent to a standard linear example.

Theorems & Definitions (24)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • Lemma 3.2: Lemma 5.1 in firstpaper
  • Proposition 3.3: Theorem 3.3 in firstpaper
  • Proposition 3.4
  • proof
  • Corollary 3.5
  • ...and 14 more