ℤ∕p × ℤ∕p actions on Sn × Sn

Fowler, Jim; Thatcher, Courtney

doi:10.2140/agt.2024.24.1841

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$\mathbb Z_{/p} \times \mathbb Z_{/p}$ actions on $S^n \times S^n$

Jim Fowler and Courtney Thatcher

Algebraic & Geometric Topology 24 (2024) 1841–1862

DOI: 10.2140/agt.2024.24.1841

Abstract

We determine the homotopy type of quotients of $S^{n} \times S^{n}$ by free actions of $ℤ_{∕ p} \times ℤ_{∕ p}$ where $2 p > n + 3$ . Much like free $ℤ_{∕ p}$ actions, they can be classified via the first $p$ –localized $k$ –invariant, but there are restrictions on the possibilities, and these restrictions are sufficient to determine every possibility in the $n = 3$ case. We use this to complete the classification of free $ℤ_{∕ p} \times ℤ_{∕ p}$ actions on $S^{3} \times S^{3}$ for $p > 3$ by reducing the problem to the simultaneous classification of pairs of binary quadratic forms. Although the restrictions are not sufficient to determine which $k$ –invariants are realizable in general, they can sometimes be used to rule out free actions by groups that contain $ℤ_{∕ p} \times ℤ_{∕ p}$ as a normal abelian subgroup.

Keywords

group actions on manifolds, Postnikov towers, binary quadratic forms

Mathematical Subject Classification

Primary: 57N65, 57S25

References

Publication

Received: 28 August 2020

Revised: 16 December 2022

Accepted: 30 March 2023

Published: 16 July 2024

Authors

Jim Fowler
Department of Mathematics The Ohio State University Columbus, OH United States
https://math.osu.edu/~fowler.291/
Courtney Thatcher
Department of Mathematics and Computer Science University of Puget Sound Tacoma, WA United States

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Volume 24, issue 4 (2024)